Discover cards are accepted in the UK at places that take Diners Club cards, but Diners Club is a pretty niche card in the UK so I'd take a guess that most places which accept DC are the same kind of places that'd also take Amex.

Discover card acceptance outside the US

Re Amex acceptance in the UK, this question often seems to get quite different responses from different people. I'd surmise this perhaps reflects the shopping and spending patterns of those responding. Whether fair or not, Amex in the UK is generally seen as an upmarket thing and so the likelihood of retailer acceptance is seen in that context.

Or, to cut a long answer short, don't count on Amex being accepted!