If you have read this thread, you would see that 'tourists' seem to do well at Selfridges
Harrods is probably the most competitive place - due to sheer footfall. Not sure if I have seen any reports of a wishlist birkin/ Kelly being granted there. I may be wrong so feel free to correct. You may have better luck at one of the other boutiques. not sure whether the wishlist system is for UK residents or open to anyone (?)