Hello Jonathan:
Last August our family of four spent three weeks in the UK, with 11 of those nights in London. The two things that we paid to get into were:
-Tower of London (and we spent an entire day there)
-Westminster Abbey
All of the other London attractions that we were interested in were free (national museums and galleries).
I looked into the London Pass but didn't go for it. Instead, we bought a 15 day Great Britain Heritage Pass and used it at many, many castles, cathedrals (including St Paul's Cathedral in London)around England. Many of our journeys out of London were day trips (we had bought rail passes).
When we went to Canterbury Cathedral (which was not covered by the British Heritage Pass), we chose to attend Evensong (by donation) at 5:30pm before heading back to London. It was amazing. After the service we were able to linger and look around (saw the place where Thomas Becket was murdered...big X on the floor).
What we found was that even with 11 nights in London (seven full days as we did day trips away), we didn't have time to see everything. We chose not to go to Buckingham Palace (too expensive and crowded) but we did make a day trip out to Hampton Court Palace which was one of our holiday highlights. We also spent lots of time just walking around taking in the sights.
I am convinced that we made the right decision regarding the passes. One day's admission to Warwick Castle for our family was GBP50 and our family pass was GBP165. I calculated that we used over GBP300 worth of admissions over the 15 days.
Best regards,
Helen