London to Bayeux and back again by rail.

Unfortunately, London and Bayeux are extremely poorly connected, unless you are a GI boarding a landing craft for an invasion!

Although it is very fair to call your post a Frequently Asked Question, it's actually a poor use of a first trip to Europe, because of the travel time overhead. Because of the train tunnel you mention, ferries are practically obsolete, and the crossing can be as rough as it ever was. Not all ferries take pedestrians, and none of them leave from the city of London. In Paris, you have to change train stations if the rail trip is all in one day.

https://community.ricksteves.com/travel-forum/general-europe/london-normandy-paris-help-with-itinerary-transport-options

https://community.ricksteves.com/travel-forum/france/d-day-normandy-visit-with-13-year-old

https://community.ricksteves.com/travel-forum/france/day-trip-from-paris-to-d-day-beaches-acf8ace3-7773-4e57-b464-b178dcd45e1f

It's a different topic, but do you know what an Open-Jaw air ticket is? If you plan an actual stay in Paris or Bayeux, you should be flying to London and home from Paris. Today, this costs not more than a round-trip to London, for most itineraries. This eliminates half of that travel-time overhead.

Be aware that Eurostar trains are priced like airline tickets. Business-day tickets are more expensive than midday tickets, and advance purchase discounts can be substantial, but they are non-changeable and non-refundable. They are also not available in the immediate weeks before the departure. It's also a factor that Eurostar currently demands a 90-minutes before departure check-in, because all the customs and immigration formalities are done at the DEPARTURE station, not after arrival in-or-out of the European Union.

You can take a ferry but it's not really for expedience.
You would take a National Express coach from London direct to Portsmouth International Port at 1100 or 1830 (2 hour journey time), then a ferry either mid afternoon or overnight (Brittany ferries to Ouistreham/Caen) then connecting coach to Caen Railway Station for a train to Bayeux.
It is easy enough but the ferry doesn't arrive at Ouistreham until 2130 (day boat) or 0730 (overnight)- so you would be arriving at Bayeux at close to midnight.
The fare is pretty similar to the train (in fact a bit more expensive), so really the train is the better option.
The only train from London to Paris is the Eurostar.
Leaving London St Pancras International at 1022 (so a very similar time) you would arrive at Paris Gare du Nord at 1347. On a sample date of 12 September that train is currently £86 each.
You then need to cross to Paris St Lazare. Currently if you booked on any of the hourly trains after 1558 they cost 19 Euro each 2nd class, 30 Euro 1st Class.
It is a 2 1/2 hour journey so the 1558 gets you to Bayeux at 1820, some 5 hours earlier than the ferry would.

Just take Eurostar from London to Gare du Nord in Paris. Then get in the taxi queue and take a cab to Gare St. Lazare.

You should buy Eurostar tickets ahead of time, and when we did this trip, we bought our tickets to Bayeux in advance as well.

I like to use Trainline. I would allow a couple of hours between trains so as not to have a nervous breakdown.

foot passenger permitted ferries are few and far between.

Brittany Ferries do run 3 trips a day from Portsmouth on the south coast of England (where much of the invasion force started from) to Caen, a short train ride from Bayeux (but the train station is not near the port) in September.

Outbound 08:15-15:00 or 14:45-21:30 or overnight 22:45-06:45
Inbound 08:30-13:15 or 16:30-21:15 or overnight 23:00-06:45

at around £96 each way for 2 on the 12th to 15th September (random date). Plus cabins (£5 for a deck seat, up to £85 for the best cabin, each way). Plus train to Portsmouth and perhaps overnight hotel. Plus taxis each end plus train in France.

The Eurostar was described above.

If you are thinking late September remember that when the Channel gets windy it gets lumpy. The D-Day invasion was not originally planned for June 6th, it was delayed because of weather in the Channel - in June.

Reading one of the above threads I wonder about a grasp of French geography- people suggest taking a ferry to Cherbourg for Bayeux, when Ouistreham is so much closer. For the same reason I didn't suggest a ferry to Le Havre (also overnight) which is pointless when Ouistreham is closer.
It is also totally untrue that ferries are "almost obsolete", other than in the minds of those who wish them to be so. They still have a thriving traffic for their healthy market segment. Pretty much all the English Channel ferries, other than the ones from Dover, still take foot passengers.
In this particular case the ferry is not the fastest or most sensible route for you.
To many, with different needs, they are the most practicable.

The real issue with this trip is the time it will take. First, your trip from London to Paris will be like flying. You have to go thru passport control in London at the Eurostar train gate, so allow plenty of time for that process. Once you arrive in Paris, you have to get from the train station (Nord) to another train station (St. Lazare), which you can take the metro line or a taxi. (20-25 min.) From St Lazare you take a train to Bayeux. Total time is a least 8hrs.

It is doable, but it is a long day. You could always stay in Paris near St. Lazare and leave the next day if you want to break up the travel time or perhaps fly into Paris, do Normandy and then move onto England. This would save you the backtracking trip to England. Enjoy the beaches, it is a great experience.

The trip is not too hectic IMO but I don’t you so maybe it is for you. Changing stations in Paris is not difficult. You will arrive on the Eurostar at Gare du Nord and will need to get to Gare Saint Lazare. I’d take a taxi from the official line outside the station or arrange a car with https://monchauffeur.sncf-connect.com/reserver-un-taxi-vtc. See this thread for lots of information about the trip, including the current location of the taxi stand: https://community.ricksteves.com/travel-forum/france/paris-gare-du-nord. The Eurostar is just a train so don’t get your hopes up for some kind of spectacular ride. Understood that it goes under the Channel and that is an engineering marvel but for passengers it is 30 minutes or so in a tunnel and the landscape when you are above ground is pretty blurry and boring between London and Paris. That being said, I think that it is you most efficient connection from London to Paris to get to Bayeux. For your costs, you need to go to https://www.eurostar.com/us-en for the Eurostar tickets and https://www.sncf-connect.com/ or https://www.thetrainline.com/en-us for your tickets between Paris and Bayeux. The taxi fare between stations should not exceed 22€.

I did exactly what you want to do last year. (At least one way.)

Take the Eurostar to Paris (Gare du Nord). Grab a taxi to Gare St. Lazare. Train to Bayeux. It's fairly simple.

For two, trying to take the Metro between Paris Stations, will include a lot of walking and stairs. It might save you 5 Euros. Get in the taxi queue and take a cab. (Ignore anyone that approaches you and asks if you need a taxi. Get in the official taxi rank.)

You will take care of French/Schengen immigration in London so when you get to Paris you just go on your way.

I took the ferry years ago before the Tunnel was open. The seas can get very rough.

It is also totally untrue that ferries are "almost obsolete", other than in the minds of those who wish them to be so.

Maybe you could tell us what that market segment is? Does it involve buying cigarettes and liquor? All you can drink buffets with chums? A tourist's time has value, and the overhead-time of the ferry is to me, a deal-breaker.

While it is very much an OPINION, I think it is a fundamental mistake to pair "the only important site in France" with a trip to England. They are too far apart. (When we stayed in Dinard, we were told it has the highest absentee U.K. home-ownership in France. But that doesn't make it "close" for a tourist with no car.) Paris is one of the great cities of the world, and it should be part of every American's, eventual, trips to Europe. That's the time to visit the D-Day beaches.

When my wife and I made our first trip to Europe together, in our youth, it was a week in London and a week in Paris. We did not go to Normandy until at least our seventh time in France (including business trips.)

I have never taken the ferry, but my wife has, and she was seasick all the way.

I've had no issue with seasickness on Channel ferries traveling between May and September, but 2 of my 5 planned ferries were canceled. All my ferry trips were into or out of St.-Malo, farther west than Caen, but I'm not sure that matters when the issue was cancellations.

For the EuroStar I've read that since Brexit, it's really smart to show up at St. Pancras 2 hours before departure time. I'm not sure what the situation is like in Paris.

I live in rural Normandie and have several friends and business associates who are English expats living in France. All of them make frequent trips "home" for various reasons, and all of them take a ferry, most often from Ouistreham near Caen. I haven't done that yet, as I haven't had a need or interest in heading to England, but ferry crossings are quite common and very popular, both among the expats living around here as well as among their friends and family members who come from England to visit them here in France.

As for the topic of this question, I would either do the Eurostar to Paris with a connection to Bayeux, or fly from London to Carpiquet airport between Caen and Bayeux, and then go to Bayeux via bus, bus/train, or car.

The OP asked about both ferry and rail routes, which I managed to do, and effectively dismiss the ferry route, without recourse to poorly informed hyperbole.
Brittany Ferries was originally formed as, and still is a co-operative of French farmers to get their produce directly to market in the UK. That very important part of their business is still expanding (even after Brexit) with new and chartered tonnage coming in to serve it. The short sea routes out of Dover etc may have had a past reputation as duty free and booze cruises. The longer Western Channel routes never really did that. To suggest they did/have is not being in possession of the facts, or being knowledgeable of the holiday and trade markets of the 4 countries BF now serve. To many these sailings (and others from as far west as Plymouth) represent a huge saving of physical time and of fuel over driving up to the 'Le Shuttle' and then back down through France. The ships, the terminals and the on-board experience of BF are very smart and very gallic.
One of the problems is that the United States domestically has few if any ferries such as exist widely in Europe, and the ferries have hugely improved over the years, from basic to almost cruise style. Yes I have been in horrendous weather on the Channel.
There may be personal reasons why the OP feels it important to go to Normandy. For all we know they may have been to Paris before, so it is not being helpful to re-direct them to Paris. Everyone has different priorities.
I recently did my first trip report, and I was scared stiff of being excoriated by forum members for my priorities and my speed of travelling, as neither apparently matches what is deemed to be correct, and the only valid method.

very interesting and loads of detail

Here's some additional information about France-England ferries, and recent experiences:

https://community.ricksteves.com/travel-forum/england/travel-canterbury-to-france-by-sea

I recently did my first trip report, and I was scared stiff of being excoriated by forum members for my priorities and my speed of travelling, as neither apparently matches what is deemed to be correct, and the only valid method.

I know the feeling. I value reliable plumbing and a good mattress more than "quaint family-owned inns with creaky floors". I will take a taxi or hire a car service when the public transportation options seem overwhelming. Once, I actually had a wonderful European vacation with a EurailPass!

The latest post from Tim refers to the short sea routes out of Dover, not the Western routes being discussed here, and which had already been effectively dismissed anyway on time and cost grounds. There is at least one other such thread in similar vein. A totally and utterly different experience. As different as chalk and cheese.
No one travelling rationally in at least the last 50 years would have taken an ex Dover sea route to reach Normandy. Even in their glory days that would have been the long way round.
The western routes still take and welcome foot passengers. Portsmouth is the total antithesis of Dover to travel through. Yes you can get to Portsmouth port by train, but it's a walk or a taxi, not sure why you would when the coach takes you to the front door of the terminal.
It would be nice if an air route still existed to Caen, sadly that went with the failure of Flybe, and the linked closure of London Southend Airport (recently re-opened)- which was/is a rail served airport.
Although uncompetitive on a time basis there is an irregular train service from Paris CDG airport to Bayeux, although the through train from Lille Europe (so avoiding the change between Paris stations) is not currently running.

One of the problems is that the United States domestically has few if any ferries such as exist widely in Europe,

The U.S. has the world's busiest passenger only ferry system carrying just over 13 million passengers a year. It runs 24 hours a day, 7 days a week, 365 days a year and rarely stops in bad weather.

And it's free.

It's called the Staten Island Ferry.

I'll admit I haven't read all the responses so far, so if this has already been said, so be it. I wouldn't count on the Ferry system for some date in the future. There are too many reports of unreliable service, strikes or cancelled crossings. I personally had a Ferry crossing cancelled last minute many years ago. Fortunately, they rerouted me on a different ferry (both departure city and destination city), and they arranged for my rental car to also be at another location. But disconcerting to say the least and I just barely made it. The other thing I will advise is to not refer to it as the "Chunnel" train. I don't know if this term was every used extensively in Europe, but it certainly isn't now. It is called by its name: The Eurostar.

My group of 4 just did the Eurostar from London to Paris and then to Bayeux return. As mentioned in other replies, do arrive at St. Pancras (and Gare Du Nord going back) 90-120 minutes early to get through the passport checks. The lines can be long. We found the Eurostar staff to be very helpful and professional. Once through you can enjoy a coffee while you wait for departure. And the transfer between Gare Du Nord and Gare St Lazare was quite simple - we actually hopped on a bus to Trocadero and got a view of the Eiffel Tower in between trains (I had allowed time between trains for this). The trip to Bayeux was beautiful - lovely country to see on the way and easy. We actually found that train to be nicer than the Eurostar. The biggest challenge was getting a taxi to our accommodation once in Bayeux. The train station is tiny. Depending on where you’re staying it may be worth arranging for a pick up ahead of time.

Bayeux is absolutely lovely - otherworldly almost. You will enjoy it. If you haven’t booked your tour - we had a personal guide named Steve with Overlord Tours - picked us up at our door and 10 hours later after an incredible personalized experience dropped us off. Highlight of the trip.