I am not fully versed in US immigration law, but I suspect Harry is still a few years away from being eligible to become a citizen. And, if he becomes a citizen, he is still required by US law to renounce his UK titles (in his case, Prince, Duke of Sussex, Earl of Dumbarton, Baron Kilkeel, and Knight Commander of the Royal Victorian Order).
I am not sure, however, if renunciation in the US affects his legal ability to use his titles in the UK. I suspect it doesn't, but, to be consistent, I would imagine Harry and, by implication, Meghan would voluntarily stop using their titles right away once Harry became a US citizen.