Oh, dear. They are likely under the impression that you were but a product of their ancestor's imagination. Not everyone is well aware of the vagaries of this place.
Cards are considered legal tender, yes - you hand them to the shopkeeper, they take the number via machine, and then return it to you. That number is a code of sorts to permit them to take the amount you owe them directly from the bank, instead of you having to trouble yourself. Perhaps if you ever had an account with a tailor or grocer, it would be that manner of thing.
However, paper money still exists, as well as coins. They are still commonly used, though more likely to pay for small things like a paper or tea. Very few people would use paper money to pay for such large bills as rent on rooms, for example.
As regards a good tailor, I'm not certain I could be of assistance there. My husband buys his clothes pre-made in stores; custom made clothing is rather expensive nowadays. Harrod's is the best store, in my opinion ... I'm forgetting my history; Harrod's might even have been around in your London.