Anyone know a bus that goes between these two?
Hi, Polly, would a website be of help to you? Sorry, I don't know the bus number but here are two ways to check: www.tfl.gov.uk, gives you bus maps of London and www.visitLondon.com is also a good resource.
This does not answer your specific question. However, when we were in London last year, the news shop where we bought our Oyster card gave us a free map of the bus routes. We found that very helpful over the next 10 days in navigating our way around the city by bus.
No direct bus route from Haymarket, but if you go to the stops at Piccadilly Circus in Piccadilly itself or Shaftesbury Avenue you can get a 19 or a 38 to the British Museum. Get off at the stop outside the Church of St George, Bloomsbury and the Museum is just on the other side of the church.
We took bus 14 from South Kensington to the British Museum. You get off the bus a couple blocks away and walk (tottenham court road). Worked great for us.