I live in London and would strongly advise against renting a car for the London leg of your trip. London is not built for driving around in cars - despite there being millions of them. It is so much easier, and faster getting around by public transport.
I’ve been to the Harry Potter Studio Tour. My son and his friend - about 9 at the time loved it. I was about 44 at the time, and loved it. We did the tour independently and the staff there were fantastic - I’d do the tour independently rather than on a guided tour.
You’ll probably find yourself in or near Victoria for large parts of the London leg. There is a coach station there and trips run regularly to Harry Potter Studios. That said, given that you’re likely to do the tour as you leave London, you could probably start renting your car from this point of your trip.
Once you’re out of London, you’ll be travelling on motorways to your other destinations. That is where the car is convenient.
In London, there is one attraction, that not many know about because it is essentially public transport, but offers incredible views. I suspect your kids will enjoy it as well.There is a gondola crossing over the Thames that offers spectacular views over London and of the O2 arena as well as planes coming in to London City Airport. I take all visitors to London there and we spend a day at the restaurants there and if you book in advance you could if you choose, also Climb the O2 - kids aged 10 and up are ok to do this.
https://www.moneysavingexpert.com/travel/london/
The above link is from a site I use for all manner of things from choice of electricity supplier to travel. Use the advice here and you will not get screwed like a typical tourist. Tip 6 gives details of the gondola over the Thames and there are other good ideas - 15, 18, 22, 28 for example.
This is a link to climbing the O2
https://www.theo2.co.uk/up-at-the-o2/climb
I hope you have a great time. And yes, when in Liverpool, you’ll definitely have to:
https://www.youtube.com/watch?v=08083BNaYcA
Edit: As pointed out by Slava, below - the museums are fantastic attractions - the Natural History Museum was a particular favourite of my son’s. It is located near a load of other museums including the Victoria and Albert and the Science Museum. All of these are free entry and at the very least a great backup plan if the weather is iffy. They are all located near South Kensington underground station. Funnily enough, I was down there yesterday accompanying my son to the Imperial College open day.