Interesting comments from the city-dwellers!
In the Lake District, I would expect to be able to use my Amex at petrol stations, major supermarkets and chain stores, some medium to higher-end hotels.
Elsewhere, at independent shops, small hotels, pubs, museums and attractions etc. I wouldn't even be sure of them accepting cards. And if they do, there might be a minimum transaction value and/ or they might accept debit cards only, not credit.
You won't get far if Amex is your only means of payment.