Daw, aren’t you a cutie anon.
Well, ‘cheers’ is basically a chummy way of saying thanks, so the right way to respond would probably be a “you’re welcome” or “come again!” Saying cheers back would be sort of like saying thank you for them saying thank you… which could make sense, but they might just think you’re mimicking them. As loath as I am to say it, there are some British colloquialisms that just don’t carry well with an American accent (one of my American friends tried to say 'bugger’ when she was visiting and caused a bus-load of people to nearly kill themselves laughing).