the british library is a lovely place. as is the british museum, which contains some fascinating typographic relics in the form of hieroglyphics and various other type related artifacts (coins, etc...)

also the type museum in stockwell but i'm not sure if thats reopened yet.

Also lovely is St Bride Library.

St. Bride is the Holy Hall for printing and typography matters. For a more wide scope of inspiration from different disciplines of the arts, *don’t* miss the Victoria & Albert Museum.

The British Museum has an impressive reading room as well. IIRC
As does the V&A. I attended a lecture about some of the books several years back. The people working there were very friendly.

Thanks for all your suggestions - looking forward to checking them out!