Ticket timing? 9:30 Westminster Abbey, what time Churchill War Rooms?

This is strictly personal and no one here really can predict how much time you or your wife would want to spend there, much less teenage boys. With my boys I would have predicted on the short end, 2 hours maximum. With some walking and time for lunch (what kind of lunch, a long sit down or a quick pickup?), I would make the War Room tickets for maybe 2:00. If you are done with lunch sooner you can do some of your sight seeing walking.

That is a lot of walking in one day. Are you trying to tire out those teens? Are you yourself and your wife up to that much walking?

Given how close the war rooms are to Westminster Abbey it seems slightly counter intuitive to go up to Buckingham Palace in between.
I wonder if that couldn't be fitted in at some other time.

The palace is the only part of the plan that seem out of place considering the other places you’re going to. It’s about a mile from the abbey so going to it, then on to the war rooms will be an extra mile or so walking. I don’t recall many places to eat near the palace. If that’s the only time you’ll have to walk by the palace, go for it. If you’ll be in town for a few days and could fit it in another time, you could save some walking. It is an easy walk.

Buckingham Palace is out of the way, so I would do the Abbey, lunch (if you all need it in between), and then the War Rooms. I would say that getting tickets for 1:30 or 2 for the War Rooms would be a good bet. You have 30 minutes to enter, so if you have 1:30 pm tickets, you have until 2 pm to enter.

I would see if you could go by the Palace later in the day or on another day.

I agree that Buckingham is too out of the way and suggest keeping the abbey and war rooms in succession.
We are doing the same thing at the beginning of June. I believe our entry for Westminster is at 10 and we plan to be at the war rooms by 2:30. My husband is a bit obsessive about the abbey and still feels that 3 to 3 1/2 hours there is sufficient (my kids, 12 and 15, have learned to gracefully go along with it). Unless you are also abbey-obsessed, you may be fine with less than 3 hours. For ease and to maximize our time, we plan to lunch at the abbey cafe. The food looks good, at least online.
I will say that the last time we saw the abbey we kept it pretty basic (something about having a toddler) and spent the next 14 years saying, "I wish we'd have seen this or that part of the abbey, or done a tour, etc." This trip is to make up for those regrets. So I do suggest being aware beforehand of what aspects of the abbey you think will interest you most, and if there is a lot on your abbey list, don't cut your time short.

We visited CWW & museum Saturday, 1 April. 10:30 timed entry tickets. It was not busy then, but by the time we finished at 1:30 the place was pretty full and getting more so. The CWW & accompanying museum is a warren of rooms. The audio guide helps immensely. Guidebook helps you remember what you experienced,or not, during your visit.

Thank you all for the input! It helps!