This building is 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". It's identified as the former Cock in the Listed Buildings Register.

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.