Not shown as a pub on any available OS map (back as far as 1885), so probably closed a long time ago. Even in 1885, the adjacent bridge is called "Cock Bridge", but the building is marked as "Bridge House".

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)