There is no arrivals lounge for any airline in Singapore.

There is one interesting option.
BA flies into Changi Terminal 1. Arriving and Departing pax mix in SIN and you have access to all airside facilities. You can pay to access the Swimming Pool at the top of Terminal 1 (next to the transit hotel) which will give you access to showers at the swimming pool. It is the recommend option for pax who wish to shower while in transit and do not have access to a lounge.

The one catch is that you will still be airside showering, while you checked luggage will be going around the carousel by itself. Not sure how long before airport staff remove the bags.