The short run west
Bournemouth is about 19 miles from Brockenhurst and takes roughly 35 minutes: A35 west, on to the A31, then the B3073 through Hurn to the airport. It is the second-closest airport to the village after Southampton.
The one thing to plan around is the A31 westbound on summer weekends, when New Forest visitor traffic is heavy through Ringwood. Outside those windows this is a quick and unremarkable journey.