Friend, you should post some photos of your rifle for a positive ID on the old girl.
The 'FTR/'53' means that she had a Factory Thorough Repair in 1953. This effectively returned the rifle to 'new' condition.
But a rifle should not be marked as being built by BSA and by SSA BOTH. They were two different plants, both in the city of Birmingham and both operating in the period of the First World War.
Your FACTORY BUILD information is located (generally) on the Butt Socket (that strap-looking thing where the butt attaches; it is part of the main Body of the rifle), just under the Bolt Knob when the bolt is closed. Flip up the Bolt and there is your build info. It should consist of
CROWN
GR
PLACE OF BUILD
ShtLE III*
DATE OF MANUFACTURE
It is possible that your rifle is a BSA original with some SSA parts; that did happen, as did the reverse. This COULD have come about during the 1953 FTR. It's okay because the critical parts would have been FITTED and now are correct on YOUR rifle.
But photos are needed.
.