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There is no requirement to carry ID in the UK. Nobody, not even the police, can require you to have ID (though they can require you to give your name, and can ask for ID if you have it). If you are driving you can be asked to produce a license, but have several days to do it.
People can require to see ID to do business if it is reasonable. If you buy a student ticket they can require proof that you are a student. An airline will require ID for security reasons. A bank will require ID before they let you cash a cheque. But nobody is going to stop you on the Underground and ask to see ID without a reason.
Whether a credit card counts as ID is going to depend on what you are doing. Airlines generally want photoID, banks may accept a credit card (especially if they can verify it isn't stolen). If you want to do something that might require ID, ask what is acceptable.