Transfer from LHR T5 to T3

Morning all,
In January I'm flying from DUB-LHR with BA then transferring to T3 for my flight to SFO with VS all on one PNR so my luggage will be checked through, I have a 2h5m layover.
In order to maximise my time in the CH am I better off following "flight connections" signs or heading straight out to the Taxi rank and Cab around to the Upper Class Wing for DTCI "a la JoeyC"
and giving my drivers details en route?
Many thanks
Dominic
In January I'm flying from DUB-LHR with BA then transferring to T3 for my flight to SFO with VS all on one PNR so my luggage will be checked through, I have a 2h5m layover.
In order to maximise my time in the CH am I better off following "flight connections" signs or heading straight out to the Taxi rank and Cab around to the Upper Class Wing for DTCI "a la JoeyC"

Many thanks
Dominic