To add to the general sense of gloom and despondency, your landing time at LHR is just that: the scheduled time of touchdown. You have to add taxi time, time to disembark, the walk/run to immigration, queuing at immigration, being asked questions at immigration, getting to the baggage hall thinking "well, at least the bags must definitely have arrived by now" to discover that someone has kindly decided to hold them back so they do not get lonely on the carousel by themselves, waiting for the bags, getting the bags, leaving through customs.....
In short, with no stacking delays, you should add an hour before you're ensconced in a cab: if all goes smoothly, you will then reach LGW with the required hour in hand. But any flight delays on the way over, air traffic stacking you above Biggin Hill, or a creash on the M25, and youve missed it.