Hi Jstroose!
I cannot help with the Churchill War Rooms but I did just book tickets for Buckingham Palace in July.
There are plenty of tickets available in August for the self-guided tours however the East Wing Highlights tour is sold out for the year.
The only other guided tour is the Garden Highlights Tour.
Buckingham Palace is both quite popular and has timed entry. If you want a specific timeslot, your best bet is to book as far in advance as you can.
To put this into perspective, the East Wing tours sold out within minutes and while the self-guided tours are not as in demand, August is tourist season.
Book early to avoid missing out.
Edited: 18 April 2024, 14:22