If I remember right, a few weeks ago that's exactly what Trump was asking his supporters to do.People could vote in person on the day and mail a vote later that same day.
UK certainly has checks for that. If you have a postal vote that fact will be recorded on the register they have at the polls. If you still want to vote in person you need to put your postal vote in at the voting station.
Also, I think there are various numbers recorded that means a check could (with some difficulty) be made after the event. That poses a theoretical risk that it can be found out who somebody voted for, but it is difficult enough and involves enough different people/records that it doesn't pose a serious risk.
I don't know how it works in the US, especially in places they mailed out postal votes even without people asking.