There was a pub at the corner of Cotton Street and Abbey Street called The Picadilly, perhaps you`re confusing the name with The Old Tontine in Gauze Street (or maybe not ... I stand to be corrected!)
Another pub long-since vanished was The Auld Hundred (The Aul` Hunner) on the opposite corner from RH Findlays.
Curiously enough, The Imperial Bar has risen from the dead a few yards from its original site at the foot of New Street, about 30 years after closing down!