First draft itinerary mom + 10/13yo sons. How do I have an empty day?

You'll probably be seeing the places on your itinerary you never had time to see on such a busy itinerary.
Every trip needs a light day at the start and at the end--to rest up from the flight and to get ready to fly out.

Well my boys want to go to London to see Churchill's War Rooms. They love anything military though. It sounds like you will have a great time!

Day 6 looks a little jammed to me. I'd allow at least two hours for the British Museum to see the highlights (Egyptian collection including mummies, Greek collection including the Elgin Marbles). They don't open till 10 AM, so you'd finish up around noon at the earliest. I haven't been to the British Library in years, but when I went I think I spent about an hour there. I assume by Harry Potter you're referring to the Leavsden Studios; I did those last summer and LOVED them, so I think you and the boys will enjoy that a lot. But, they take up a lot of time. I think we spent three and a half hours in the studios and could have spent more, plus the travel time on either end. I don't know how late they are open or what time your tickets are, but basically you may want to shift something off that day onto your empty day.

Otherwise, I agree with the poster who recommended the Churchill War Rooms, I think they are one of the most interesting museums in London! Some other ideas: take the boys to Borough Market for lunch, have fun exploring all the different options and everyone can pick out what they want to eat; get tickets for a football (soccer) game; day trip to Oxford or Cambridge; follow one of Rick's walking tours for whichever area sounds interesting (you can also download them as audio tours on his app; I enjoyed his City of London tour a lot last time I was in London).

"Day 7 (Thurs): Stratford-Upon-Avon"
You could spend the night in Stratford-Upon-Avon, then next day
(Day 8) go to Warwick Castle, as you mentioned. That is an excellent plan for your extra day.
You can get there by train, bus or taxi from Stratford-Upon-Avon. Then train back to London.
We did this as part of our May 2016 trip, and loved Stratford-Upon-Avon and Warwick Castle.

A few additions and deletions or reconsiderations for the "first in line kinda family.".

Day 2 (Sat) 8am Coffee and a nosh at Gail's Artisan Bakery on Portobello Road while the vendors set up. Then out the door to experience Portobello Market. From there to Kensington Palace then biking and a picnic in Hyde Park. Forgo the Natural History Museum.

Day 4 ( Mon ) Drop Westminster Abbey

Day 5 (Tues) After your visit to The Tower of London, add walking up to the glass covered Leaden Hall Market, to let the boys find the entrance to the Leaky Cauldron. From there visit Brick Lane and Spitalfields Market. Lots of eye candy. Far more intriguing to a 10 and 13 year old than St Paul's. After lunch if no one is afraid of heights experience Tower Bridge this way http://www.towerbridge.org.uk/plan/ instead of merely walking across it. Then the HMS Belfast.

Day 7 ( Thurs ) Imperial War Museum, London Eye, jet boat ride on the Thames.

Day 8 ( Friday) Westminster Abbey, Churchill War Rooms

I love Claudia's suggestions and agree with her comments. Both the Imperial War Museum and the Churchill War Rooms are worth it, particularly if your boys know at least a little about WWI and WWII.

However, if your boys are Harry fans, then I would at least check to see if you can get tickets to the Harry Potter studio tour in Leavesden during one of the days of your trip. See https://www.wbstudiotour.co.uk/ It is absolutely worth it and is a fun day trip--I substitute it for Stratford-Upon-Avon, unless your boys are already Shakespeare fans (and if so at their ages, hurrah--this drama teacher salutes you!).

Also, if walking after the long flight from the US seems daunting, take one of the open-top bus tours. It was a great way to orient our family of four (two boys) to London. We capped our first day of with a genuine pub meal. :)

I may have missed it but are you there in time for Buckingham Palace?? I am there in September and got my tickets.
You an also use the "empty" day to go back to places you want to revisit.

If you really want to stay busy..
There's Abbey Road (Beatles) you can recreate, or visit the Sky Garden https://skygarden.london/sky-pod-bar (free) or enjoy Afternoon Tea. If you've never been to Paris, there's a day trip via the chunnel. There's also plenty of markets to check out. http://www.visitlondon.com/things-to-do/shopping/market/londons-top-markets#RcwwerTXG5uE1Koc.97