My first thought would be to check with the Sofitel (attached to T5) to see if their 24-hour fitness center and spa is accessible on a day basis:
http://www.sofitel.com/gb/hotel-6214...hrow/spa.shtml
To reach the hotel from T1, you would take the Heathrow Express from T1 to T5. There is no charge for this short ride (in either direction).