So with the routing I have on a 125- ticket number should I be able to UUA the BA legs?

Yesterday I thought not because TA tickets can only be upgraded if there aren't any other carriers on the ticket.

However, the travel agent tells me that it should be possible (if BA weren't being awkward about the nature of the ticket as I described above) and as a preferred sales agency they do this every day.

I'm really confused and would appreciate some guidance from those in the know.