Many timetables (but NOT as far as I can see details of the Westbury to Swindon service) are already posted at
http://firstgreatwestern.co.uk/Content.aspx?id=281If you look at the Paddington to Cheltenham, Bristol, S Wales tables there are a number of footnote "M" entries that say "To / From Westbury" and they've to the Chippenham and Swindon times. On the Tuesday to Friday ...
Chippenham at 06:10, 07:30, 14:20, 17:50 and 22:06
Get to Swindon at 06:50, 08:20, 15:00, 18:40 and 23:01
From Swindon at 06:40, 08:40, 14:23, 17:43 and 22:10
To Chippenham at 07:20, 09:30, 15:03, 18:33 and 22:50
Extrapollating normal buses subsitutions, add 15 minutes for Melksham, 30 minutes for Trowbridge and 45 minutes for Westbury to Chippenham times. So (for example) 06:45 from Westbury, 07:00 Trowbridge, 07:15 Melksham, 07:30 Chippenham gets into Swindon at 08:20.
Sign at statation? Probably a few days before it starts on recent past (FGW) form.