I would pull up the reservation on United.com and look for the ticket number. Does it start with 001? AA should be able to help. They should have a record of the BA ticket they replaced and be able to reticket you in business class, assuming space is still available. If the ticket number starts with 016 then you're kind of hosed. AA has washed its hands of the whole matter and there is no way UA is going to move them to business class based simply on a recitation of the events up that point.

The time to have sorted this was when the agent was rebooking them. You need to ask at that time what they are doing and push back if they are downgrading you or rebooking days away or giving you a ridiculous routing, And in order to do that effectively you need the tools to research flight options yourself.