For regular London Taxi's I use the GETT app. But you can just hail one on the street, usually. There's Uber too, but I prefer a traditional taxi.
As others said; don't book any tours in London except for maybe some London Walks walking tours. All of the things you mention you can do on your own.
No need to book ahead except for the Churchill War Rooms. Many museums, such as the British Museum, are free; you just walk in when you want to go.
Out of town, you do need to book Harry Potter well in advance.
Why do so many visitors to London think they must visit Stonehenge? Do you have a particular interest in this? If not, skip it. It's a whole day on the bus to see ancient stones.
Go to Windsor and/or Hampton Court, these are fairly easy daytrips close to London.
Take the boat (no need to book, it's just like taking the bus) to Greenwich, it's a pleasant half day.
No need to stay in Victoria, but the location of the Comfort Inn there is fine. Book it if the price is right for you, London hotels are expensive and will not get any cheaper if you wait.

Honestly don't look at 'must see' lists. There are so many posts on this forum from people wanting to do Stonehenge/Bath/Cotswolds or some other place far from London.
If you want to see a castle, Hampton Court is fantastic, with so much history. You can even get there by boat from London. And Windsor is a lovely village, close to London, and the Queen actually lived there. Warwick Castle is 3 hours on a bus one way. Do you want to do that in July when there is so much to do in London?
What do you and your family want to do and see? Do the granddaughters want some time to themselves, maybe go to Portobello Road or Camden Market?

Hampton Court is a palace rather than a castle and Windsor is not exactly a village and does have a castle.

Ya know . . . the Tower of London is a castle - I guess because it is called 'The Tower' some don't realize it is.

Agree with Tulips re Hampton Court Palace with a caveat about transport. IMO/IME is better to take the train TO Hampton Court and then if the weather is nice take the boat BACK into central London. The river trip is lovely but takes quite a bit of time so taking it to the palace means you can't get there until early afternoon. Taking the train gets you to the palace in the morning before the biggest crowds and then after visiting you can take a leisurely mid afternoon boat ride back.

Tours
 

I have used a website called Viator for day trips with much success. It could be helpful for your trips out of the city. Viator is an aggregator of tours, somewhat like hotels on is for lodging.

I live in London for half the year and to get around I nearly always use either the tube or buses. Cheaper and quicker than taxis for most journeys. You also get to see more of the city from the top deck of a bus.

There are apps for black cabs - Gett and TaxiApp I think. You can hail cabs on the street but you may find it difficult if raining or in the rush hour. On the odd occasion I need a car I generally use Uber or Bolt, though Uber is unreliable in certain areas.

I wouldn’t bother with Warwick Castle, it is a long way for a pretty average castle and I think you would find it a disappointment compared with the other sights on your list.

Our London trip is getting close - June 30th. What is the best city map for London?

Google map on your phone! Maybe a big paper map to show your kids in the hotel room but everyone uses their phone now to navigate.

No, not everyone.
I still like having a Streetwise in my bag.

https://cimg6.ibsrv.net/gimg/www.fod...f171461a5.jpeg

I always used a A to Z ....

I agree with nyse. I have lots of Streetwise maps for various cities. For myself, I think it's much easier than looking at a small screen, especially if you want to see where something is located compared to something else.

i have the same Street Wise, a similar Art Wise, which are both good, and my favorite - a Pop Out that folds up smaller than the streetwise. It will fit in any pocket. I've carried one or another of these for decades - My old Pop Out finally wore out last week and I've just ordered a replacement.

If you are happy using googlemaps then that would be the way to go. Far, far quicker to find your route from A to B whether by bus, tube or foot. it will also provide a fairly realistic idea of journey times, delays etc. The TFL app is also worth downloading. If you really want a paper map then A to Z.

Agree the phone is fine some, but we really like a paper map too.

Exactly. Gives you a better overall perspective.