Westminster Abbey was extremely crowded when I arrived at noon a week ago. I had read here that it's more crowded in the morning... I had to work hard to see everything before the 4:30 PM closing time. YMMV; I am not at all religious but very interested in the decorative elements in grand churches. Note that after you leave the nave, there are still things to see on the way to the exit. You should not plan to rush through the Abbey because it will not be physically possible to do so. Until around 3 or 3:30 PM or so (when it did get better) it looked like the Vatican Museums in terms of packed bodies.
I spent over half a day at the Churchill War Rooms a few years ago (pre-COVID). It was the Churchill Museum that took most of the time. You need to buy this (time-specific) ticket ahead of time, because the ticket-buying line can be significant and the CWR sometimes sell out. This is a problem with your schedule, because it's difficult to know when you'll be ready to leave the Abbey.
The V&A is a magnificent and huge museum. It accommodates large numbers of visitors relatively well (much better than the British Museum), but the jewelry exhibit will be very crowded unless you arrive at the museum at opening time and go straight to the jewelry section--which doesn't fit into your plans. Jewelry is small, so you have to be able to get very close to display cases to see the itmes and read about them. If this isn't a strong interest, I wouldn't target this area for a mid-day visit. The V&A building isn't easy to navigate. You often have to take substantial detours to get from Point A to Point B. Examining the floorplan online ahead of time to pick out the areas of most interest and figure out a logical route will be helpful. Check the website before your visit for announced closures of specific sections and adjust your plans accordingly.
The first floor of the British Museum is packed until rather late in the afternoon. The second floor isn't quite as bad but will probably still seem crowded except by comparison to the first floor. As with the V&A, it takes multiple days to see the whole thing. Check the museum's website and identify your top priorities. I've taken London Walks' British Museum tour and it is fine, but there's nothing the tour guide can do about the crowds. I don't know what the presence of the Bayeux Tapestry will do to visitor flow at the museum. For others with impending visits: I recommend a Friday visit because the museum stays open until 8 PM (they do start clearing people out before then), and the last few hours tend to be much, much less crowded. However, in my experience there is no food available after about 3:30 PM, even on Friday.
Three days (or 3-1/2 days) isn't long for London. You will certainly want to return. I'd recommend taking the time to fully enjoy the places you choose to visit rather than rushing through them and perhaps feeling you need to go back to the same places on your next trip.