The one exception, if a TA booking, is Upgrade Using Avios but that's only by one cabin, so it sounds like WT to WTP here. This you do by calling BA, but only if it's a BA ticket, BA metal and not a complicated ticket.

You are able to up purchase to CW or even First when within 24 hours of departure of the first leg by calling up, as well as a fare difference at the airport, or if available, AUP (see Dashboard thread).