Acceptance of Discover (along with various other cards from other parts of the world) has only just started in the U.K. I have seen that Marks & Spencer advertise that they accept Discover, but suspect that most other outlets wouldn't bother to advertise the fact. Most places use the same machine for all cards, and rely on the machine to confirm that the card has been accepted. However, you may find that some places are wary of cards they have not seen before, and they may refuse to accept them because the commission charged may be higher. That is the problem with the acceptance of American Express.
You may also find card readers which only accept the chip & pin cards which are now the standard in most developed countries except the United States. There are usually ways around that, and most hotels, restaurants, etc, especially in London, will handle that without even commenting.