Westminster Abbey
The Abbey charges tourists for entry, but not for worshippers. Attend a church service for free and enjoy some of the finest choral music in London from the choir. Choral Evensong at 3pm (Sa Su) or 5pm (weekdays), depending on time of year, is an especially good bet. The Abbey is

The Abbey charges tourists for entry, but not for worshippers. Attend a church service for free and enjoy some of the finest choral music in London from the choir. Choral Evensong at 3pm (Sa Su) or 5pm (weekdays), depending on time of year, is an especially good bet. The Abbey is
Check the official site before travelling; hours and prices can change.
More museums

Sir John Soane's Museum
Free, tiny, deranged — Soane's own house preserved exactly as he left it in 1837.

The Wallace Collection
Free national museum in a Manchester Square townhouse — armour, Boucher, the Laughing Cavalier.

The British Museum
Free, vast, civilisationally heavy — go for one room, not all of them.
Source: wikivoyage