If their credit card is a Visa, Mastercard, or American Express, they shouldn't have any products buying anything online. I still use my Visas and Mastercards from the US, and have never had any problems... it's all one system.
The only thing I can think of that *might* be a problem is if by "UK-based" they mean that the credit card has to have a billing address in the UK... but that would be very weird. Just tell them to give it a try and to let you know if they have any problems.