Leon,
AFAIK the BBMF is funded in the same was as HMS VICTORY and the National Army Museum. Each Service is permitted to fund one unit from public monies.
I am not sure your time was "free". Service personnel are funded 24hrs per day 7 days a week; Not Required for Duty is a bit different to not being paid. This is not to take away from the fact that you volunteered your time on a very worthwhile cause.
RNHF gets/got a small grant from the RN of about £240K pa, but this may have stopped. The CO of the unit is also now a serving aircrew officer.