How’s this London itinerary?

Yes: doable! :-)

On my phone, so I can’t scroll back and answer in order of your post, BUT:

I recommend Blacklock for Sunday roast. Not only is the food truly excellent, but it’s a short stroll from the National Gallery.
https://theblacklock.com/restaurants/blacklock-soho/

Scheduling only Hampton Court and the National Gallery, plus a meal, into a single day is smart. Some would argue that you will have to give the National Gallery short shrift after spending the bulk of the day at Hampton Court, but I think you can do both. After all, there is only so much magnificent history, art, and architecture your eyeballs can take in in a day! :-)

I’d prioritize Evensong at St. Paul’s over Buckingham Palace, no contest.

You haven’t scheduled much time for the War Rooms. If you’re a history buff, you might feel rushed/cheated.

Good idea to include plenty of time for the Tower. It’s incredible! As for pubs in the area? All I can say is to avoid The Minories, which is a quick walk from the tube and looks adorable, but is a big let-down.

Thanks for the reply! How long would you allow for the War Rooms? We can forgo the London Walks tour at Westminster Abbey and just go earlier to allow more time for the War Rooms.

I adore Westminster Abbey and could spend hours there. That said, my husband is a huge WWII buff and had to be dragged out of the War Rooms after four hours.

If you are into WWII history, I think 3-4 hours would be more like it. :-)

This sounds doable, I would just note two things:

-I would have Sunday roast near Hampton Court Palace. Hampton Court Palace is very big, and door to door it will take about an hour to get to National Gallery, so you'd be having lunch quite late otherwise.
-I would prioritise British Museum over Buckingham Palace.

Saturday: It depends on where the London Walk ends. Which one is it? If it is headed toward St Paul's that would possibly work. I don't think you could do the Buckingham Palace Tour afterwards as sometimes the walks run a little long.

Sunday: I think your Sunday is too much. For myself, Hampton Court Palace takes a full day including going and coming. I am not much of a Sunday Roast person but this meal may be more geared to midday? Others will correct that for me and let you know if this is something you could get when you return to London by mid-late afternoon. I think the National Gallery on top of Hampton Court is too much to expect. I'd go with your priorities here.....Hampton Court Palace vs National Gallery and something else.

Monday: Depends on how much of a Royal Fan you are. I've done both and would vote for British Museum depending on the make up of your group. The War Rooms I would say would be a minimum of 2 hours but as others mention, it takes me 3-4 hours. If you skip the rooms about Churchill and just walk thru the area where the War Rooms are that will shave some time off. There are lots of little stations where you can listen to oral histories from the folks who worked here - men and women who had kept the secret for 50 years. Even their families did not know where they worked. I will swear that I can smell Churchill's cigar smoke when I am down in there.

I am a fairly determined London visitor although I've been many times. I am more of a detail visitor the more times I go. I think one of the hardest things to determine is how long it will take to get from place to place without causing yourself undue anxiety.

It’s not clear whether you’re planning Sunday roast before or after the National Gallery - but it’s a lunchtime/early afternoon thing. Most pubs tend to serve it from 12, possibly till as late as 5pm but really 1pm to 3pm is probably the optimum time.

So I agree with the suggestion to have it near Hampton Court Palace if that works for you.

As others have said, Sunday Roast is a lunchtime/early afternoon meal rather than an evening meal. Perhaps get to Hampton Court at 10 am and then have a Sunday Roast nearby at around 3 pm? If you do go down this route, I'd aim to book a table at a nearby pub (I am not a Sunday Roast person - too much food, too early in the day - and don't really know the area around HCP so am unable to recommend anywhere).

After this, you probably would not be able to make the National Gallery or any other London museum before they close (also you may be too blown out).