It's not quite as simple as a British/American distinction. In British English bathroom means bathroom. Every home in the country has a bathroom. A bathroom is a room with a bath in it. It's what we call the room where we wash, bathe, shower and so on. The regional difference lies in the fact that American English, to the confusion of the rest of the world, also uses the word 'bathroom' for a room that doesn't have a bath in it at all. Speakers of American English also use 'bathroom' to mean 'toilet', as in 'I need to use the bathroom'. 'Restroom' is used in the same way in US English. 'Washroom' is sometimes used in British English for the facility in a school, hospital, office building or other work premises where toilets and washbasins are found. However, it is not an everyday word in British English. It tends to be used in names of companies ( e.g. ABC washroom supplies) or by architects, facilities management departments and so on. Nobody in Britain would ever hear a classmate or colleague say 'I'm going to the washroom'. So, in that respect, it is not an equivalent term to the American use of 'bathroom.'

Washroom is not used in Britain to mean bathroom or toilet. The word is sometimes used for commercial enterprises - we have a company near here called The Washroom. It is a laundry.

Washroom is British usage and bathroom is American usage. They mean the same thing.