Hello, I was wondering if there are mortarman jobs in the British army?

Yes you can but you do have to serve as a general infantry soldier first. The timeline for this is dependent on your approach and desire to become a mortar soldier. Essentially be honest about your aspirations and attend the first mortar cadre you can and it may only take a couple of years.

Also, is it possible for a citizen of the United States to join the British army?

Unfortnately not - unless you are a dual passport holder. You must be a British Citizen, a British subject under the Nationality Act of 1981.