Economy of rail travel with a family...

I have traveled with one child and still rent a car most of the time...I like flexibility and spontaneity.
When traveling with friends 3 or 4 it is definitely cheaper for us to have a car. Some people just don't like to drive on vacation but I love roadside picnics, taking any side road that looks interesting, not worrying about train schedules and being able to stop for any photo opportunity...I am a photographer, so this is the kind of travel I love...

Add up the train tickets and compare.

A car that will hold four people without an lot of junk will run about thirty bucks a day. Gas was averaging 1.30 ppl last week - - maybe eight bucks a gallon. You should get close to forty miles per. Parking is extra, and can get steep.

Driving is just about impossible in London: you can walk most places faster, good luck on finding parking near anything, and the congestion fee is a pain. York isn't really that horrible, just find a place to park and hoof it. Bath sucks worst of all.

When I was lugging three rascals, we always drove -- but it was never a city-to-city operation and, even in London, you could park down by the Tower, get finished and stuff them in the back, and drive up to Westminster and find another place to park with maybe one trip around the block.

It's the stops and deviations in the middle of your three listed places that's going to make the difference. if there are none, choke the money and ride the trains.

Check out some of the special offers for rail in the UK. There area family discount cards, group travel cards, etc., plus you can see major savings when you make advance purchases (usually about 12 weeks prior to travel date). We always used a BritRail pass, but I understand they are not the bargain they used to be. We really enjoyed being able to be flexible about travel dates, destinations for day trips, etc. We could make changes based on weather, etc. To us it was worth it even if it wasn't the cheapest. We also liked being able to watch the scenery without worrying about directions, etc. It was time to relax, listen to music, sketch, etc. And our kids knew they had to keep up with 'their' stuff, too. We always made time just prior to our destination to gather everything, check the seats, etc., and move towards the doors. Parking and petrol (gasoline) is VERY expensive and we just didn't want to have to worry with it- or have to plan routes, deal with traffic delays, etc.

You don't mention ages of the children, JoAnne, but be advised that children under 5 are free, children between 5 and 16 are half price or less, and children from 16 to adult are full price.

Using Groupsave, groups of 3 or more save one third. No Railcard required.

Using Two Together Railcards (£30 for a year) a couple traveling together can save 1/3, too.

There are ways to get the cost down.

Advance Tickets can be very inexpensive.

driving in london is AWFUL!!! get a car on the outskirts or elsewhere. . . try to imagine sitting on the wrong side of the car driving a stick, and driving on the wrong side with a million and one roundabouts in a very busy and crowded city. it was pure insanity. (add to that a broken GPS, we didnt know it was broke until it said we were in Bath and we were in some other town) once outside of the city it wasnt bad, 4 lane highway, even better. but then you have to find somewhere to park. . . for us it would have been worth the bit extra to get on a train. we traveled by train with kids from london to york, hampton court, and edinburgh. really nice trains and if you know your schedule far enough out the tickets arent bad priced.

What places do you plan to visit? I honestly think that whether you choose car or train (or both) depends on your itinerary.

For example, London to York is only 2 hours by train and at least double that by car.

We have traveled with our son in Europe from an early age and we almost always make sure we have a car. The exception would be if we are going a long distance to move to a new place (more than a 7 hour drive or so) or if we will primarily be in a large city with good public transport. It is not worth it to have a car in most really big European cities like London or Paris. That said, it is very nice to have a car to get to places outside the city...for example, you may want to go out to Fountains Abbey near York and a car is the best way to get there. York has some fantastic remote parking lots at the edge of town where you can park a car and ride a shuttle in to the city center for minimal cost. Two days in York is plenty so you won't spend much to park and you'll have the car to move on to a new place and see all of the wonderful things in between!
It is SO nice to just stash stuff in the car as you would do driving around at home. It's entirely worth it to me to not have to navigate public transport and be tied to a train or bus schedule that might not allow us to move around at our own pace. It's also nice to not have to whittle down what you want to take with you for the day because you're worried about carrying it around with you. I'm talking about things like coats and umbrellas that are bulky and a hassle to haul around, not valuable things like iPads or cameras that you wouldn't want to leave in a car.
Navigating is so incredibly easy now with all the GPS options you can use...no plotting required. Just punch in the destination and follow the directions on the screen.

Thanks for all your input. I do think a car will be the best bet for the bulk of our time once we leave London. I am dithering a bit on our arrival though...considering going straight up to York from Heathrow (into King's Cross I guess) and then coming back to London all by train; doing London (without a car), then picking up a car after that for 5 days or so of touring (Bath, Cotswolds, maybe Oxford etc).

That came up when we thought about our arrival day...transatlantic red-eye, morning arrival, and tired people. My husband did a drive across Ireland post-red-eye one year and it was...let's say "stressful". ;-) He's willing to grab a rental then and there, but I think driving to York would be a bit much at that point.

Or maybe we get the car at Heathrow and do the southern part of our loop first, stopping somewhere closer than York the first night...Bath area maybe. Keep the car through the trip to York and do London last. That red-eye leaves you arriving at a time when you can't really check into your accommodations, but with luggage to manage. Depending on where we end up, we could drop the luggage at our hotel and then head out, but I prefer the idea of staying in travel mode until we can actually check in, and freshen up before we start exploring. ...But with a car you have that private space built in, to lay down your burden physically and metaphorically, even if you're still moving!

So, clearly, I'm still all over the place!! :-)

Out into the countryside around Bath, yes a car is useful, but in Bath itself a car is as useful as a chocolate teapot. Expensive and difficult to park, and driving is difficult in the town. Before driving in and around Bath be comfortable with driving (a non-automatic transmission?) on hills.

"...we thought about our arrival day...transatlantic red-eye, morning arrival, and tired people. My husband did a drive across Ireland post-red-eye one year and it was...let's say "stressful". He's willing to grab a rental then and there, but I think driving to York would be a bit much at that point."

"Or maybe we get the car at Heathrow and do the southern part of our loop first, stopping somewhere closer than York the first night...Bath area maybe....That red-eye leaves you arriving at a time when you can't really check into your accommodations, but with luggage to manage."

Are you comfortable getting off the red-eye and getting straight into a car and driving?

One alternative is to check into a Heathrow hotel when you get in, leave your bags there. I suggest the Hotel Ibis on Bath Rd., right off the north runway. ($74 per night)

You could take the short, very short, bus ride to Hampton Court from the hotel. Spend several hours at Hampton Court, bus back to hotel, you're in for the night to eat supper and crash early into bed. And you have not wasted an afternoon; you've seen one of the sights on your itinerary.

Next morning, check out of hotel, get rental car, head out well-rested and not very likely to have an accident. You may also have a clearer head about where you want to go first in the car. And you may enjoy the drive more, now that you're not all bleary-eyed and tired (including the driver).

Just one option.

Was just reading another thread, "The Roads From Bath To York", and one poster who lives in England had this to say:

Posted by kbr61263 09/24/14 11:56 AM
"along with Oxford and London, Bath and York are the worst places in England to drive (and I am British and have lived in both areas).......If you do it in 5 hours (Bath to York) the roads wont be scenic, its a route I know very well, I live near York and my parents near Bath."

To read his entire post, here's the thread:
https://community.ricksteves.com/travel-forum/england/the-roads-from-bath-to-york
It's the very last post on that thread.

I have no comment on this one way or the other. I have not driven in Bath or York; have not driven from one to the other. I do rely on the opinions of those who live in England and have lots of experience driving in these areas, however.

Don't drive after getting off a red eye flight, we don't want another of these:
http://news.bbc.co.uk/1/hi/england/west_yorkshire/7767693.stm

Here is an article from Trip Advisor outlining the driving rules/laws for the UK for visitors:
"United Kingdom: Driving Tips For Visitors"
http://www.tripadvisor.com/Travel-g186216-c9626/United-Kingdom:Driving.Tips.For.Visitors.html

When I first decided to rent a car in England years ago, I first hopped into a cab and paid the driver for about 45 minutes to drive me around, get to know the feel of driving on the other side, how to turn corners, navigate roundabouts, a brief explanation of the different paint patterns on the road, lighted signals, etc., and asked him about driving laws I wasn't clear on. Best expenditure of pounds I have ever made!
I have driven in just about every city you list except London. Driving is not that bad if you are patient and take your time. I've done a lot of driving in Oxford, Bath and York and it really is not too difficult. Just go slow and have someone else to help navigate and you'll be fine.

Being a transportation professional, I encourage at least part of the journey by rail just for the experience. If your children have always lived in NS, long distance rail may be something they haven't had much opportunity to try. The walk from York station into town is short and pedestrian-friendly. The station itself is historically interesting. A vote for mixing up the modes for some added adventure!

See this thread currently going on, and read the last 4 posts made by Karen (from Santa Rosa), Pamela, MC and Nigel:

https://community.ricksteves.com/travel-forum/england/car-travel-in-uk

Very eye-opening as to what it's really like to drive in the UK with all of the roundabouts.

One more comment on driving on the "wrong" side of the road (from the US perspective) that I noticed and remembered today. In the US the yellow line is always on the left indicating that it either the far left lane on a freeway, the divider line on a 2 two line or divided highway with no median. In the UK all lines are white. So a 2 lane road looks identical to the right side of a dual carriage way, unless you have a car coming straight at you in the right lane. The friends we stayed with made the comment that they hear about head on collisions due to this exact reason.

Again, don't drive in UK straight off the plane.

Anita's comment is spot on. We toured with in our friends car for a day around Fife and Stirling. I sat in the passenger seat, and that was fairly bizarre. but the experience helped me understand roundabouts, signs, etc to help with the navigating. It's very disconcerting as a passenger sitting in the "driver's" seat but with zero control. I would say I was very jittery. Likewise, it took us 2-3 days before we actually walked to the correct door on the car. The first morning, we both actually sat in our seats first before realizing we were both sitting on the wrong side of the car. Worth a few laughs.