Yes, we very much get old American shows, particularly your westerns. We also take a lot of your Kids TV as well (we have the whole of Nickelodeon and Cartoon Network) among other things. We also get the big obvious ones like HIMYM, Big Bang Theory and GOT. If the network thinks it will sell, then we get it.
Nom, it isn't just the BBC that sends you programming from the UK. Who wants to be Millionaire was adapted for US audiences from ITV for example.