If I am actually visiting a state that sells Mega Millions, is citizenship relevant at the shop? I know online sales across borders are a different issue. Would be useful to see purchase, claim and tax questions separated.
Yes. Mega Millions says visitors to the United States may buy from an American lottery retailer and do not need to be a US resident or citizen to win. The player still has to meet the age rule and any other rule of the jurisdiction where the ticket is sold.
A US Mega Millions ticket remains subject to the selling lottery's validation and claim process, and a prize is handled in that jurisdiction. Mega Millions also says its tickets are not sold outside the United States, so a UK website should not be described as an official Mega Millions seller. Identification, US tax and moving prize money to the UK are separate questions.