grid reference TL 974 343 (approximate location)
The Cock is also listed in Bear Street, so probably stood somewhere near where Bear Street and Birch Street meet. It's understood to have been open in the 17th century.
Cock fighting (usually in a pit) was probably a regular feature near here (typically organised on Shrove Tuesday and other holidays), and hence the name was widespread until it was banned in 1795.
(Most pub, location & historic details collated by Nigel, Tony or Keith - original sources are credited)