If the flying doesn't work out....
Fly to LHR. Railair coach to Reading station from LHR. Train to Chippenham. Coaches provided from Chippenham to Badminton.
www.badminton-horse.co.uk
but if it does....
Also, from BRS to Badminton... Coach from BRS to Bristol Temple Meads station (fairly frequent) train from Temple Meads to Chippenham.
Hope this is useful.
sB