Lol, despite the OP's repeated assertion that camping is not an option it continues to crop up! Do people actually read the thread?
To the OP, whilst I understand your reluctance for camping are you aware that camping in Europe is often very different to that experienced in North America? European campsites usually contain swimming pools, playgrounds, bars, entertainment etc and the accommodation is not confined to tents, there are all sorts of accommodation including some quite luxurious cabins.
Whilst good weather cannot be guaranteed in the UK, August is usually reliable, last summer was a particularly hot one. How about some time in London, the Science Museum is excellent and very geared towards children of all ages with lots of interactive activities. It's also next door to the equally good Natural History Museum. Both are free and of course there's the wealth of other attractions in London.
After some time in London you could head towards the South Coast. There's a fantastic small theme park, Paulton's Park https://paultonspark.co.uk, which is ideal for children aged the same as yours. My kids are a bit older now and they prefer the thrill of the big theme parks which is a shame as Paulton's Park was a guilty pleasure of mine. Paulton's Park is close to the New Forest which is an incredible national park, full of wild ponies, free ranging cattle, pigs, sheep and donkeys and contains a plethora of wonderful villages including Brokenhurst, voted the most beautiful village in England. There are plenty of places to stay and some real foodie gems to encounter. I'm sure the kids would love the New Forest Water Park www.newforestwaterpark.co.uk/activities/aqua-park.
Further west along the coast is Lulworth Cove www.lulworth.com/visit/places-to-visit/lulworth-cove. This is an ideal place to visit in the summer and the sheltered, shallow cove provides excellent safe swimming. It's also a great place to explore with some incredible natural features.
You can then continue westwards along the Jurassic Coast and go hunting for fossils, they are literally just hanging out of the cliffs and along the beach, especially after a storm and in the highly unlikely event that you don't find any there are a multitude of shops in all the towns and villages along the coast that sell them.
From Dorset you could continue west into Devon and then Cornwall or alternatively, head north and perhaps hit Minehead, a traditional English seaside resort www.visitsomerset.co.uk/explore-somerset/minehead-p421683 and from there on to Bath and perhaps The Cotswolds if you're interested in their popularity.
Of course there are other routes to take out of London that will encompass other fantastic areas of Britain but I'll leave it to others to make suggestions.