Timing questions

When we were there some people took the bus from Salisbury to see Old Sarum. If you're budgeting your day be sure to include a generous helping of transportation time (or take a taxi)

For your last day, and the others for that matter, you might want to consider London Walks. Excellent tours including both the streets of London and tours in some of the London museums.

Only going to answer the first question.

The area around Westminster Abbey is Parliament Square. M-F opening hours at the Abbey are 9:30 to 4:30am. Peruse the website to be certain the Abbey will be open to the general public during your dates. https://www.westminster-abbey.org/

Have you looked at the Parliament website? Asking because I enjoyed a tour of Parliament and having afternoon tea there. This was a few years ago. Not sure if it’s still feasible after Covid lockdown. Worth some research though. https://www.parliament.uk/visiting/visiting-and-tours/

As far as lunch options between Westminster Abbey
and the Churchill War Rooms, consider a pub lunch.
Adam and Eve or Two Chairman, The Sanctuary House or the Red Lion all serve meals. All are within easy walking from both the War Rooms and Westminster Abbey.

OR a grab a sandwich and crisps or salad to take away from the Pret a Manger on Tothill Street. From there walk over to St James Park. Find a bench and people watch. Afterwards follow Birdcage Walk up to Buckingham Palace. If the Royal Standard is flying The King is in residence.

1 - I would book for 2PM for the War Rooms. In addition to the food possibilities Claudia listed, there is a cafe at Westminster Abbey called the Cellarium. I’ve not eaten there in a number of years but the location is very cool! I won’t get it on this trip either as the Abbey is currently closed to ready it for the Coronation.

https://www.westminster-abbey.org/visit-us/food-drink

2 - Are you planning to take the shuttle bus out to the stones from Salisbury? It runs a loop and stops at Old Sarum on the way back from Stonehenge. I’d give it an hour. The shuttle driver will point out the schedule of times they stop there. If you’re energetic I’ve also walked out to Old Sarum from Salisbury’s City Centre.

3 - Tate Britain and National Gallery don’t need timed entries so you can decide at the time what sounds good. I don’t know about reservations for the Eye or the photography exhibition so no advice!

Have a wonderful time!

Getting the boat from Tate Modern is a good idea. You could go to Greenwich in the opposite direction if you wanted to do something other than a gallery visit.

Don't pre-book the London Eye. You don't want to be there at a fixed time and it's raining or the visibility is poor.

For your last day....Of the things you mentioned, the National Gallery would be first on my list.
If you go to the Natural History Museum to see the photography exhibit, I would also walk a short distance away (just down the street) to see the Victoria and Albert Museum (The V&A).