As OP is flying into T5 and out from T3.. I Suggest a Day room at Hilton T4.
Hilton Bus from T5 to Hotel..check-out around 6pm I expect for Day Room unless can negotiate later.
Then Train from T4 to T3 for Spa and Dinner in GF
Sorry no access to CCR or Cabanas.
My other suggestion is Shower and Breakfast in T5 Arrivals, then Bus from T5 to Windsor for Castle Tour and Afternoon Tea.
http://www.londonbusroutes.net/times/071Heathrow.htm Or
http://www.londonbusroutes.net/times/077Heathrow.htm Back to T5/Train to T3 for Spa and Dinner.