Looks like taking the train from Heathrow Terminal 5 to Cheltenham Spa requires a change of trains at London Paddington Station. I'm thinking of taking a taxi (about 29 pounds?) to Slough station and catching the train from Slough to Cheltenham Spa. I can't tell from the train website: Does anyone know if it is a direct trip with no transfers? Thanks!
Consider taking the National Express coach from the Heathrow Central bus station (close to terminal 3)to Cheltenham. It will set you down right in the centre of Cheltenham at the bus station, whereas the train station is away from the centre. It takes about 2-2.5 hours. Check the National Express website for times, reservations etc.
Slough to Cheltenham Spa involves a change of trains at Reading. There is an express coach service from Heathrow to Reading station for £13 - see http://www.railair.com/ As Lesley suggests, National Express run a direct coach from Heathrow T1-3 to Cheltenham town centre. They are not very frequent and it takes a minimum of 3 hours 5 minutes, up to 3 hours 40 minutes. See www.nationalexpress.com
From Terminal 5 you can get to Reading quicker on average via the Railair Coach than using the Heathrow Express and train. The trains to Cheltenham from Reading only run on a two-hourly schedule, but there is an additional service each hour that can be connected into at Swindon. Incidentally there is a bus from Terminal 5 to Slough which takes 25 minutes, but this would still require the change at Reading.