There is to be a new Coach Service from Basingstoke to Heathrow from 21 August- #730 and 731.
Change at Basingstoke for Southampton, Salisbury and beyond.
This is in addition to the Rail Air coach to Woking. The journey time is faster than to Woking, and Basingstoke is closer to Southampton. It takes just 1 1/4 hours and is run by Newbury Buses.
For the first two weeks the fare will be £1, then £20 each way, or £25 period return (valid for 1 month).
This is the timetable- https://images.reading-buses.co.uk/downloads/730-731-web_0.pdf
This is a very useful new service.