Mr87, the ballot finishes on the 21st, but I am not sure how long it would take them to count up the votes and verify them.
I would expect that the result would take some days to be announced because based on the previous legal goings on Unite are going to be extremely careful to check and double check before announcing.
They do have to give 7 days notice of a strike, so it is extremely close, assuming of course that you dont believe that BA will operate 100% Long Haul services as they have stated.
I have a flight going long haul just before you, and at this point im not worried.