According to a 2006 report, Britain's most expensive public toilets are near Parliament Square costing 50p. This inspired me to find free toilets near to the Houses of Parliament.
These toilets are only free to use if you also use the Methodist Central Hall cafe on the lower ground floor, which is open from Monday to Friday 8am-4pm and Saturday to Sunday 9am-4pm. The cafe has reasonable prices for the area and serves full English Breakfast, sandwiches, salads plus hot lunches, desserts, cakes, etc.
It's an interesting building and you can contact the Visitor Services department for guided tours.
Check the location map. Basic directions are to walk past Westminster Abbey, away from the Houses of Parliament, and cross the road where you can see the Methodist Central Hall and the Queen Elizabeth II Conference Centre. Go in the main entrance of the Methodist Central Hall and turn right to go downstairs to the toilets and cafe.