L&M describe this covered exchange for merchants as being on Corn Street, and this National Archives article suggests similar:
In the 14th century the records [of the Corporation of Bristol] were kept in the Guildhall and it would appear that with certain exceptions there they remained until the 16th century. After the dissolution of the chantries, St George’s Chapel adjoining the Guildhall was closed and the building was then used as a repository for the city archives. About the same time (1551) a new Council House or Tolzey was built in Corn Street where provision was made for keeping some of the archives, although it seems that most of the records remained in St George’s Chapel. The Town Clerk must have kept the important administrative records in his custody.
However, this article suggests the Tolzey was just north of All Saints’ Church.
The map points to the current location of the merchants’ “nails”, which are on Corn Street.