If you can wheel and deal your way into the BA Arrivals lounge at T4 using BA Gold status or BA long-haul business/first travel, then you can also get the spa treatment -- since once into the Arrivals lounge, the Spa personnel don't look at your boarding pass/card.
On the departure lounge side, the Spa has more vigorous gatekeepers and you will need a boarding pass (and perhaps a BA Gold card) to avail yourself to the spa treatments which are only available to long-haul J/F passengers and BA Gold members themselves.
[Are QF business/first long-haul travellers still permitted to use the Arrivals and Departure facilities for spa treatments too? Perhaps some more regular QF fliers could share their experiences.]