What train would do that????

I've been connecting between LHR and LGW quite regularly recently and I always use the National Express Coaches which take approximately 50 Minutes Airport to Airport... They also seem to understand traffic quite well as they would of course normally take the motorway, but on one trip, the driver virtually didn't use the Motorway at all.

A taxi may shave off a few minutes... or not (depending on traffic and knowledge about congestion), so I don't think there's any quicker way, is there?

I'd suggest to use NX - that way your ticket will state your (scheduled) departure and arrival time. If you miss your connection due to your flight coming in late (assuming it's on one PNR), BA should accomodate you... Confirm with their hotline what their suggested way of connecting is.

Update:
In Detail

Book the NX Coach departing at 9:05 - arriving at 9:55. That gives you 1 hour for arrival and baggage reclaim and a further 40 minutes for delays and checking in at LGW - perfectly doable IMHO, have a little faith.

Note that they'll let you use earlier coaches if you are outside prior to the 9:05 departure.