Seeing this post several days later, but having just returned from London last week, it appears that most every museum has made great efforts to provide suitable dining spaces and places for afternoon tea. Even St. Martin's in the Field church has a surprisingly delightful, very affordable dining/tea space in their crypt (underneath the church, but pay your respects to the folks buried under the tombstones in the floor).
Our visit to the tearoom at the fabulous Courtald Gallery, in a shady outside corner of the place a few days ago, however, was chaotic. Maybe a lot of regular staffers were out sick, or new trainees were left in charge of the place, but they brought out food on trays and than asked people whether or not they'd ordered it, then moved on to the next table, etc., unsure of whom had ordered what. They were left with one unclaimed plate of scones and then determined it didn't belong to anyone, but they weren't going to make it available to us, either, so they'd technically run out of scones for the afternoon...and tarts, although those were still being touted to potential customers, who were then suddenly disappointed. Then they didn't get a table bussed that had lots of uneaten salmon sandwiches left on it, which an opportunistic pigeon quickly turned into a huge mess.
Then trying to pay was a problem, as the server couldn't get the credit card machine to work, but we also had cash and got that resolved.
As far as not drinking tea at Tea, lots of warm and cold beverages are usually on offer, including coffee and lemonade. And have you ever had chai (which in many languages simply is the word for "tea") but in English-speaking countries is tea flavored with cinnamon, cardamom, sugar, and other spices that, with milk, is an entirely different drink! The tearoom at the Courtald did get the chai right!