In the US you will more commonly hear "mailman". "Postman" is understood, but sounds old-fashioned. As James pointed out, we also use the gender-neutral terms "letter carrier" or "mail carrier". You''ll also hear "postal worker" but that more often refers to the people who work in the post office itself, not the people delivering the mail to your house.

As far as I know, 'mailman' is used in the US and 'postman / postie' in GB for a man/ woman whose duties involve sorting mail, letters and small packets in the district sorting office and then delivering these to their respective addresses.

Never heard of mailman. In the UK we use postman or postie.

In the United States, people try NOT to use either term because it has the word "man." It hurts the feelings of the women who deliver the mail. So the term that we try to use is "letter carrier."