Opinion needed: Bath/Wales or York/Edinburgh

Only speaking to the Bath vs. York comparison, I took my daughter to both on one trip - she was a little older than your oldest. York had way more stuff for us to do. There was better walking via The Shambles, there was the York Castle Museum, which was unbelievably interesting. Walking the walls, Railway Museum.

In Bath we did the baths, a day trip to the Cotswolds and Stonehenge I think. But York won out . We did Edinburgh on a different trip when she was little - maybe 11 or so. There was a ton there for her and we were there during the Fringe Festival so we saw some cool shows. (Parenting mistake: saw The Full Monty, front row with her - she still discusses my poor decision making).

Never been to Wales so can't weigh in.

You don’t say whether you are renting a car or using public transport. Be also aware that you can fly between Paris & Bristol (which has shuttle buses to Bath) as well as fly from Bristol to Edinburgh. See www.skyscanner.net - where you can also check for trans-Atlantic flights between your home and Paris / Edinburgh as well as London. (Some trans-Atlantic flights also go into Manchester = nearest airport for N. Wales).

Note that north Wales is more about scenery whilst York/Edinburgh is the built environment. If you have a car, why not consider Dorset - which is only about 1 hour south of Bath and has historic houses/gardens and a fab coast. It can be reached by direct slow train from Bath to Weymouth - but car is best for getting about.
http://visit-dorset.org.uk
https://jurassiccoast.org
https://www.visitwiltshire.co.uk/salisbury

Trains in Britain = www.nationalrail.co.uk (long distance routes can be expensive for pay on the day - best to pre-book specific trains about 11 weeks ahead).

I'm a big fan of Bath, went when I was 10 many moons ago and remember enjoying the Roman Baths and the Royal Crescent. The architecture is beautiful if you like that type of thing. I think your trip around Great Britain would be most successful if you hired a car. About 20 minutes from Bath is Longleat safari park and house which will appeal to kids of that age and will fill a whole day (lots of animals to see, play areas, maze, historic house).. 19 days is a long time to go non stop siteseeing so you might consider spending a short break at Centre Parc's at Longleat. They offer breaks in self catering chalets for either Friday to Monday or Monday to Friday. There are lots of activities on site and an awesome pool complex but you can come and go from the site so could do you trip to Bath easily from there and maybe visit the safari park on the day you leave before travelling on to South Wales. Also in the vicinity of Bath is th National Trust owned Dyrham Park if you like stately homes. Also worth a mention is Bristol which is close to Bath. You can take a pirate walking tour around Bristol Harbour and visit the SS Great Britain, take in the sites of the iconic Clifton Suspension Bridge, visit Bristol Zoo or @Bristol which is a hands on science type museum (much smaller than London science museum though).
You mention North West Wales, but I would give South Wales serious consideration. Overall there is more to do even if the mountains aren't as large. You could spend 2 nights in Cardiff, visit places likebSt Fagans folk museum (free entry but very interesting and engaging for children plus some lovely Italianate gardens to walk around), Cardiff castle, Castell Coch, Caerphilly Castle, Big Pit for Trip down into a coal mine (free entry). If they want to climb a mountain head up to the Brecon Beacons and walk up Pen Y Fan - might be a bit too much for the 5 year old but the older two should be able to manage it and there are great views. You can find great waterfalls in the Neath Valley. If you're travelling along the M4 and need a nice break stop for a couple of hours Margam Park near Bridgend is worth a visit (car park charge only) - nice place for the kids to run around and blow off some steam.
West Wales (Pembrokeshire) really is a gem. In mid June you could really luck out with some great weather and still have a beautiful beach to yourselves. Take a trip to Tenby, lovely to just wander around and you can catch a boat over to Caldey Island for something a bit different. Visit Pembroke Castle and St Davids Cathedral. Oakwood Theme Park will have something for all your kids, Folly Farm will particularly appeal to the younger two. Bluestone is a popular holiday village that you could base yourself at or there are plenty of B&Bs. You could even try 'glamping' and sleep in a yurt!. Plenty of holiday cottages but most will want to rent out by the week.

Assuming you fly into either London or Paris. Why not do London first then tour around Uk. You could then fly to Paris from a number of cities in the UK which will be cheaper than the large airports. There are budget flights from Cardiff to Paris for example. The earlier you book the better price you will get. You can then either fly back home from Paris or if you need to return to London try the Eurostar train.
At the end of June is the Football World Cup. You may experience the French and English going a bit potty during that period! Enjoy your trip.

I like Bath, but it's a bit overly touristy and over-rated IMO. It would be appreciated more by the adults than your children.

Although there are things to do in south Wales, such as the fantastic Big Pit, the surrounding towns and villages are depressing, being former coal mining towns that have fallen on hard times. The castles and the countryside in north Wales are nicer IMO.

North Wales could be combined with Chester, rather than Bath, which would be a more straight forward journey. Chester Zoo is worth a visit.

By preference would be fly into Manchester and take the train to York for 3 nights. Explore the fascinating Railway Museum and the city centre. Hire a car and drive to Whitby on the coast one day and explore its Frankenstein connections plus eat the best fish and chips in the country.

En route to Edinburgh, drive up the Northumberland coast for 3 nights and visit Alnwick (Harry Potter) and if the tide permits, Holy Island. Children love watching the tide cover the causeway. Hadrian's Wall is another possibility here.

Drop the car on arrival in Edinburgh as you won't need it there.

If you choose Bath, I agree that you should spend some time in the Cardiff(south Wales) area. St. Fagan's is amazing!

Thank you for the great ideas! Yes, we would be renting a car for touring around the country, but would be returning it before going back to London. Depending which route we took, we would drop it in a city (like York for example), and take the train back in.

With all due respect to you and your family, you might want to consider this change in your plans. Each of the itineraries you've described offers many learning opportunities for your oldest children to share with their parents - unforgettable experiences. The youngest child likely would change the dynamics due to less stamina for all-day sightseeing, unfamiliar food, and other very typical needs of a 5 year old that the 8 and 10 year old have grown through. You and your 3 children will have other trips.

We did a 28 day driving tour of South Wales and England last October. It was wonderful.

We did not go to Scotland, had been there twice in the last four years.

Bath and South Wales were wonderful, but so was York, the Yorkshire Moors, Pickering and Whitby.

We rented a car and that is another experience driving on the left, but it gives you far more opportunities to see the countryside.

Here is my review on cruise critic that includes the 28 day trip.
28 days in Britain and Celebrity Eclipse home
https://www.cruisecritic.com/memberreviews/memberreview.cfm?EntryID=599139

Before I say anything , a caveat if you will allow - my apologies , I have not thoroughly read the previous posts , but having read your query , I would opt for York and Edinburgh . With three little ones , I think that choice presents the best opportunity to give them the best introduction to what is on offer . I have visited both choices on multiple occasions and would be hard pressed to make a choice , but the children are the deciding factor for me in this instance .

If you are thinking of driving with kids all the way from Bath to Edinburgh - that is a lot of driving - much of it on boring motorways (interstates). As others have said, south Wales is an interesting possibility - as long as you avoid the miserable former coal mining area - which is found in a narrow band roughly between the M4 and A465 and west of the A4043. (You may still wish to visit the underground Big Pit mine at Blaenavon & Caerphilly Castle). It only takes 30 minutes to shoot up the A470 expressway through the former coal mining area from Cardiff to the Brecon Beacons National Park.

www.visitwales.co.uk