I spoke to soemone who said he'd done a touch and go at Gatwick on Xmas day - ATC asked him to donate something to charity in lieu of a landing fee.
This was a while ago though, things may well be different now.
Presumably as the airport owners BAA are quite entitled to prevent anyone landing at Heathrow, and I guess that could include shooting an approach maybe, but a transit should be OK? If you could get SVFR it would be fun crossing Heathrow.
And Gatwick's in class D isn't it so you wouldn't even need SVFR for that?