If I live in the UK and I buy a iPhone 7 plus refurbished from USA apple store will it work on a three sim card ?

If its unlocked, it should work yes.

Note, that the warranty will only be valid in the U.S, and you would need to physically return the iPhone to the U.S for service. They don't take international shipments for service.

Also the U.S. store will not ship overseas. So you will need to have it shipped to a physical address in the U.S, and then have it shipped to you. Apple will not directly ship it to the U.K.