Whether you have problems or not depends on where you are and what kind of shops you like to go to.
In central London where I live and work, there is no problem, it is accepted as widely as any other card. I think only my barber and butcher do not take Amex, but they both prefer cash anyway. In my daily routine of the pubs, supermarkets, restaurants, coffee shops, and the local yoga club that I frequent, I find they are all very happy to accept Amex.
When I visit my parents who live in a village by the sea, one restaurant and one pub (a chain) accepts Amex, the other shops, pubs and restaurants do not. The chain supermarkets nationwide do accept Amex, except for the "discounters" like Aldi and Lidl.
When I lived in York, a city in the north of England, it was essentially only restaurants who accepted amex, except for chain stores.
Outside of major city centres, you tend to find that Amex acceptance is reserved to either relatively high affluence shops and restaurants or to nationwide chains who have significant bargaining power.
I would advise you to bring some cash if you have no access to a Visa card or a MasterCard.