I'm not BA crew, but do work for an airline, and I assume staff travel is similar.
Jump seats are often used, but there are rules to follow. Only "non-required" jump seats, ie those that crew with a door responsility don't use, can be used for staff travel purposes. There is a minimum age, and anyone using the seats must be instructed on the use of the belt, how to operate a door (if it is a seat by a door) and various other things. No alcohol.
At "my airline", at retirement and with a certain amount of years service, there is a very reduced staff travel benefit.
It is indeed the Captain who has the final say, but certainly our FSM has input. There are very specific jumpseats which can be used, and a limited number on each flight.
I don't use staff travel, the thought of not knowing if I'm going on holiday, or being sat on a jump seat for X hours, doesn't appeal.