There is a BA staff member at the entrance to T5C Flight Connections, it is the first person you encounter before the security line. They can issue boarding passes if necessary. If you can get BA.com or the App to at least do the check-in stage it will help a bit, even if you can't get the boarding pass to the passenger. The staff member can find the passenger on FLY most quickly if the passenger gives the flight number and the first three letter of their surname. T5C Flight Connections is usually so quiet that it won't matter if it takes a few minutes to do this, they get very few people there.

Note that the staff will need the flight to be showing as T5C at their end, so typically within a couple of hours of departure, otherwise your husband has to go to the main building and back again. If the connection is long, there are so few facilities at T5C that this isn't a bad thing. But the Flight Connection staff can probably still do the boarding pass even if your husband ends up in T5A.