Apparently there are two flights a week from Brize Norton RAF to Ascension and Falklands.
These flights are operated primarily to support the British Garrison in the Falklands. The operator, Air Tanker, is the same Company that is under Contract to supply the RAF's Military-registered A330 air-to-air refuelling tankers (those operating the Falklands "airbridge" being on the civil register).
I can't see that operation being amended for the benefit of St Helena, although I was struck by the number of "Saints" working as contractors at the Falklands garrison when I visited several years ago.