Two countries and transport

fly in to one country, travel to the other and fly from there

that's the most efficient approach

Without knowing where in the Uk you’d really like to go or where you’ve been before, this request is a bit tough

But, what about, fly into Gatwick then train to Brighton and bus to stay somewhere in the S Downs National park which has some good hiking (Seven Sisters, Beachy Head), Afterwards take the ferry from Newhaven to Dieppe (I haven’t checked that they take foot passengers but am guessing they do), then see Normandy and then travel on to Parts and fly back from there.

Another option if you're interested in DDay history is to head down to Portsmouth and visit the excellent DDay museum there, which explains the whole preparation and staging process involving the invasion armada from ports along the south coast of England, then maybe hop on the ferry over to Ouistreham (port of Caen) in order to visit the Normandy sites, eventually catching the train back into Paris for your flight home.
An open jaw (multi-destination) flight plan of flying into London and home from Paris would streamline your itinerary and likely save a day or two of back-tracking I think.

Yes the Newhaven route still takes foot passengers, even if the trains no longer stop at the side of the ship in Dieppe as they used to. Now you have to go to Dieppe Ville station for the train, but much easier than at Calais.
The old Dieppe boat trains were some of the most comfortable train cars SNCF owned- absolutely wonderful to step off the boat at around 3am and straight onto the waiting train- literally armchairs you sank into in compartment style train cars. Close the curtains, switch off the compartment lights and sleep through to Paris.
Thanks for the memories.
Now it's just normal regional multiple units.

I'll share my personal experience. Flight from USA to Europe, including LHR, round-trip is always cheaper. For open jaw, ie USA to LHR, CDG to USA, flights are somehow more expensive. If you use award ticket, that maybe okay depending on whether you can grab one. Note that department tax from LHR is higher as comparing to other hubs in Europe. Fly USA to LHR, spend time in London where public transport is super easy and convenient, train Eurostar to Paris, or ferry Dover to Calais, trains in France, fly CDG to USA.

Just to chat about the public transport part. This is completely doable if you are willing to take day tours once you are at Normandy. We found Overlord tours out of Bayeux excellent. The further you purchase train tickets ahead, especially foe Eurostar, the better the prices.

How lomg are you allowing for this?

What sort of things are you hoping to do and see?

Take a day tour of Normandy.
Don't miss some of the great smaller cities and towns in the UK:
York
Cambridge
Winchester
Bath
Stratford Upon Avon
Salisbury
Canterbury
Windsor Castle

We flew into London spent a.week.with some.day trips such as York, Royston, Hampton etc as this . was our fifth trip so we had already see many of the London sites. Then Eurostar to Paris and directly took.a train to Bayeau for a few days, back to Paris for a week and home.from.CDG. the open jaw flight was not more.expensive then US to London and home. from London. Plus we did not have to waste time going back to London from.Paris.we did buy the Eurostar and Paris to Bayeau tickets ahead of time as it was cheaper that way. We did not do that for the London day trips but did travel off peak. We wanted a !ot of flexibility there. It was a great trip.hope you enjoy.

Without a car, the only practical way to see the Normandy beaches is to book a private tour. No stations within walking distance of the beaches.

You should seriously consider the no car plan if you are visiting Normandy and your "beaches" comment refers to the Battle of Normandy in WWII. The D-day sites are quite spread out, and not at all served by public transportation. So unless you want to hire a guide with car, and limit your visit to only that, you won't see much of the D-day area.

As for getting to/from England and Normandy, you either take the Eurostar to Paris and train out from there, or you take your chances on the ferry. It often doesn't run for a variety of reasons (weather, strikes, etc.). I am planning on taking the Ferry from Portsmouth to Cherbourg next June (renting my car in Cherbourg), but I also bought a 1st class Eurostar ticket as an emergency back-up. The 1st class tickets are fully refundable, so if all goes well I won't need it and it won't cost me anything. The only other time I tried to take the ferry (Poole to Cherbourg), the ferry was cancelled. I was able to get on another ferry (Portsmouth to Ouistreham) and re-arrange my rental car for pickup there. But I only barely made it from Poole to Portsmouth.

The D-day sites are quite spread out, and not at all served by public transportation.

It depends when you go in the spring. But from mid May to the end of September in 2024 and 2025 there HAS been a shuttle bus (The D-Day line) between the various beaches, with integrated connections into other bus services from Caen, Ouistreham and Bayeux.

I wouldn't count on that shuttle to appreciate the D-day area. It isn't as frequent as say a city metro, and there are soooo many sites to see that are not "the beaches." Frankly, the actual beaches themselves can be a minor attraction in comparison to many other sites.

One point about "open jaw" flights that may not be obvious: These flights must be booked as a single itinerary for the package. Use a multi-destination search function, widely available. They may save money in the long run and certainly make better use of your vacation time.

Also, taking an organized tour of D-Day locations is less about the transportation and more about the information the guide provides. The battlefields need explanations.

I agree with both of Southam's points above. Open-jaw ("multi-city") flights should cost little or no more than round trips, especially considering the time and cost of backtracking -- keeping in mind that for most of us the most valuable commodity is time. And some of the D-Day tours, like Overlord, have very good reputations. Not having to drive, plus all the info a guide can share, would make this a good value.

Thanks to everyone for all this good advice. We will indeed be booking a tour of the beaches and we will probably book an open jaw airline ticket. The info
was all much appreciated. Bea