Cotswolds and ....

My wife and I stayed two nights in Chipping Campden on our 2013 trip and found it very appealing. We stayed at Cornerways B&B and the hosts were extremely gracious. There is a good local walking trail right in the town. I know you were looking for an apartment, but B&B culture is part of the experience in the U.K., so please consider it. Our upstairs room at Cornerways was huge and could accommodate a family. We visited multiple other towns and Bourton-on the-Water was another town that we enjoyed. As the name implies, flowing water within the town, and there were ducks which might amuse the kids.

From a base in the Cotswolds it would be possible to take a day-trip to Oxford, which I found an interesting city. The Pitt-Rivers Museum has all sorts of oddities brought back to England from Asia, Africa, Oceania, etc. I think your two oldest would enjoy some time there, though it is not specifically a children's museum. Ask at the shop to borrow a flashlight; some of the displays are rather dark.

School kids will all be on vacation in the last week of May so I would make the trip in June to avoid paying a premium for accommodation and getting caught up in traffic snarl ups. Check out Centre Parc's - your kids will absolutely love it and if you spend a midweek break (best value) at any one of their locations you can still use it as a base to visit surrounding areas. Longleat is great for visiting Longleat house and safari park as well as Bath.
The Cotswolds is lovely but its appeal is in its beauty and I'm not sure the kids would appreciate it as much but if you have 2-3 weeks you should probably include it. Rather than staying in a quaint village or town you could look to stay in the Cotswold water park area and then explore the wider area with your car. Being an island us brits are rather obsessed with coastline So perhaps add in a stay in a lovely coastal area. Cornwall won't yet be in peak season or West Wales is a delight. Both areas would have nice distractions for the kids as well. Norfolk has some lovely coastline and could be combined with a Centre Parc's stay at Elvedon. Personally I would do something like 5 days Cotswolds, 5 days Centre Parc's Longleat, 3 days Cardiff, 5 days West Wales, 2 nights Windsor including a day at Lego land before flying back out of Heathrow.

We did a 28 day drive tour of S. Wales and England in October. If you have a rental car, I recommend the Volunteer Inn in Chipping Campden. Rates are reasonable and there is an excellent Indian Restaurants as well as English Pub there. Parking is not a problem.
From there we visited nearly all the villages and towns in the Cotswolds as well as Oxford, Blenheim Palace and Stratford Upon Avon. Next, we stayed one night in Warwick to see Warwick Castle, which would be great for your kids.

Here is my review with details:
28 days in Britain and Celebrity Eclipse home
https://www.cruisecritic.com/memberreviews/memberreview.cfm?EntryID=599139

Research Lyme Regis. Spent a few days there last Novmber and loved it. From there easy to explore Dartmoor National Park.

I think you should make your base in Bourton-on-the-Water. The River Windrush runs through the village. The river is not deep, children love walking and splashing in the water, it’s only inches deep. If you google children playing in River Windrush in Bourton-on-the-Water you will see lot of pictures of children having fun. Even a lot of adults sit on the side with their feet in the water.

I agree with Robin Z that the water feature in Bourton-on-the-Water probably makes it a good choice with children.

The River Windrush - further downstream joins the Thames - which of course flows through London.