LHR won't be a problem. You say "American airline operated british" - validate whether it really is AA metal under a BA flight number or vice versa. That may impact which terminal they depart from.

But either way, as long as they have printouts of everything, BA agents/employees will be able to nudge them in the right direction and there will be well-marked flight connection signs. Beyond recognizing those and the departure board, they won't need too much in the way of English skills.

For arrival in Miami, they should have all of their printouts (including return flights) with them for CBP. Having this on paper (as opposed to just in a phone) is helpful. Email them a note ahead of time explaining, in English, their trip, its purpose, your information, etc. Have them print that and attach it to their itinerary. I assume you will be outside immigration with your phone when they arrive...