Hi Oliver,
We get no priority or special handling over other traffic at LHR other than any "slot swapping" which the company may be able to organise with other own-company a/c (this obviously affects no other non-BA traffic). Even this practice is being restricted now since it can load up atc.
Having said that, we are very rarely tighter for fuel than any other long-range operation. If indications are that holding is likely at LHR inbound from JFK (e.g. strong winds, unservicabilities etc.) we can usually load about 45' holding fuel without a problem. Inbound from Barbados is a little tighter, but using the same rules as everyone else for diversion fuel can still result in 20 to 30 minutes of holding capability most times.
Bottom line - we fly by the same rules as anyone else, and if that means we go to LGW then that we do. Holding is a part of life at LHR and we plan accordingly - the fiction about Concorde's fuel capability is far more dramatic than the reality - and anyway we certainly are not in a position to require other air traffic to give way unless the company can shuffle its own assets around with the slot-swap procedure (and this is very rare - I've done it myself once in 3 years of Concorde flying).
Same at JFK although rarely a problem since we arrive there at low-traffic periods.
Best,
NW1