Not sure if I'd risk staff travel as your ticket would only be a standby, wouldn't it? Depends how time sensitive the document is I guess.
Unless the documents are BA/VS related, in which case they'd probably prioritise you on the flight.
2 hours sounds enough to me at GND, and any inbound delay is likely to lead to an outbound delay. The aircraft is likely to need 2 hours on the ground if it is the same machine inbound and out.
If I were you, I'd check in for both flights in the UK before you leave - and save any potential queuing time. Just have some answers/paperwork ready for GND and LGW immigration/customs in case they wonder why you are spending such a short time in GND.
Airline staff and engineers regularly do this sort of work all over the world and I can't see that you'd have any major issues.