I suspect you'll be glad to have pre-purchased tickets at many of those places, but the only one I've been to recently and can comment on from personal experienced is the Churchill War Rooms. Those are timed tickets, and if you just show up, empty-handed, you may be in a long ticket-purchasing line and end up with a ticket for several hours later. That would mess up your well-calibrated plans, so definitely get the CWR tickets ahead of time. I wouldn't be surprised if some days sell out, as well, though I haven't heard that it happens far in advance. The thing about a compressed sightseeing schedule like yours is that if just a few timed entry periods in a row are already sold out when you're ready to buy tickets, it can mean you have to shuffle the day's plans. That's beyond awkward if you have already bought some of the other tickets for that day!
The other thing to know about the CWR is that it's a two-part sight. The war rooms themselves don't take terribly long to see. They're rather small and crowded. You'll want to get the audioguide (unless they now have something you can download to your phones; always carry headphones with you). The time-consuming part of the CWR is the Churchill Museum. That's worth several hours for folks with a serious interest, so some thought ahead of time about how your mixed-age group will handle that may pay off. The CWR tickets are quite expensive if you're just going to see the war rooms section and then leave.
All the places you list for Day 2 are on the same (western) side of the river except the London Eye.
Many folks on the forum have highly recommended the Verger's Tour at Westminster Abbey. I'm not positive, but I think it's covered by the (not cheap) admission fee to the Abbey. I think the deal is that you sign up once you're inside.
Similarly, the Beefeater's Tour at the Tower of London draws very positive comments. The usual suggestion is to go to the Tower first thing in the morning and head straight to the crown jewels, which tends to get crowded pretty quickly, then take the Beefeater's Tour or see the rest of the place on your own.
The British Museum will be massively crowded, at least on the ground floor where a lot of the famous stuff is located--the Egyptian Collection, the Elgin Marbles, the Rosetta Stone. A pre-booked entry time will be very smart. I've usually been in line for about 30 minutes when I showed up right after the museum opened. The pre-booked entry time (unless it's at opening time or perhaps the second entry slot) will reduce the standing-in-line time (you must still go through a security check) but will not do much to protect you from the crowds. Be sure there's no one in your family who won't be able to cope with the mosh-pit environment. For what it's worth, I've been told by museum staffers the museum is busiest on weekends, Fridays, holidays and rainy days. I have personally experienced the rainy day phenomenon. This does not mean the museum is uncrowded at other times. I've been to hundreds of museums, and three stand out for being mob scenes: the British Museum, the Vatican Museums, and the Picasso Museum in Barcelona (though I think I may have been especially unlucky at the Picasso). At least you aren't paying the equivalent of $20 each to go to the British Museum.
Edited to add: Sara's post reminded me that the British Museum is open late Friday night. It does get quieter inside by late afternoon, but there's no food available inside the building late in the day, so that's as issue that requires some pre-planning.
