As a U.S. speaker (but not a teacher), I feel that it makes very little difference. I don't think it matters. Therefore, since your teacher obviously wants you to listen to British English, that's what you should do. I DON'T think it will give you any special help understanding U.S. English, but I don't think it will do any harm. The big problem, in my opinion, is finding songs with lyrics that resemble straightforward, narrative English, using sentences that tell stories and make some kind of logical sense. Some of the Beatles songs fall in that category: "When I'm 64," "Norwegian Wood," "Eleanor Rigby" come to mind. For a slight unusual one: on one of their early albums they have a rendition of "Till There Was You," and I recommend it. It was written in the U.S. as a show tune for the musical "The Music Man," but they are definitely giving it British pronunciation. Or Liverpudlian pronunciation. "I've Grown Accustomed To Her Face," Rex Harrison.

There are a couple of issues with this advice that come to mind right off. Most British rock and pop bands sing in a very neutral English so there wouldn't be much difference anyway. For instance, I used to see The Cure when they came to town every year back in the 80s and it made me laugh that the singing was so clear but when the singer said things between the songs we could barely understand him because his accent was so thick. Another thing is that a lot of the older famous British rock bands were famous for imitating (and stealing from) blues artists from the American South; examples would be bands like the Rolling Stones and Led Zeppelin, both of whom would affect southern accents while singing that were pretty far from reality. So you might end up with an earful of something other than British English by listening to bands like them. Lastly, when I think of "British" music, I think of bands like old Kinks and newer bands like the Arctic Monkeys. Bands that I love, but the English they are singing in is filled with slang that is or was common at that time and are specifically English, perhaps not good for learning. For instance, one of my favorite Arctic Monkeys songs is called "Mardy Bum" which is a slang term for a complainer in England apparently. I had to look it up before I understood the song.

Of course English knows many more dialects/accents than just 'British' and 'American' English. Generally speaking 'British' English is found more difficult to understand, due to the sometimes heavy accent. However, there exist heavy accents in American English as well. What I mean to say; it is really up to you. Everyone is different, some people have an easier time with the British accent, others with the American accent. What and who do you understand without problem? It's a question for you to answer, not your teacher.