In a word, no.
Credit card companies make their money by allowing you to borrow from them. But if you pay the balance off in full every month by the due date, you’ll never be charged for borrowing.
However, if you’re late, you will have to pay interest on the outstanding amount AND if you haven’t paid the minimum amount for that month, you’ll be charged a late fee which is typically £12.
Some cards have an annual fee that you need to pay in order to get the card in the first place, but other than that, provided you pay the full balance on time every month, having a credit card doesn’t cost you anything.