When I was in KSA (Riyadh), I knew several people that were given 'trans London connections'.
The local Saudi based travel agents, most have never been further west than Jeddah, believed that 3 hours was ample time to move between LHR getting off the flight & LGW doors closing, early morning, Rush Hours. Not normally so.
I regularly advised people to try to check-in through London, and pick up a boarding card in Riyadh for the LGW-Southern USA/Caribbean routes. This will create a little time and increase your chances of success for the connection.
Having said all this, most people arrived at LGW airside as the flight was being called, needless to say, I never heard of anyone's bags arriving across the pond the same time as the pax.
If you are with BA, they may have a connecting bus, if you are an EEC passport holder, it will increase your connection chances as I have heard of non EEC passport holders (NZ nationals) getting tied up for over 1 hr in immigration cues at LHR.
The last hurdle is our favourite road, the M25, guaranteed to throw up traffic jams and accidents when you least need them.
Good luck with the most tiring part of your trip, the 'trans London connection'.
