Now its written as Payford (as in Payford Bridge, Payford Mill ), rather than the way its recorded in 1851 census.

http://www.old-maps.co.uk/
Co-ords 374435,229328

This maybe the same building, although the local Directory for 1870's suggests a new school at Pool Hill.

The Old School House, Payford, Redmarley,GLOUCESTER. UK. GL19 3HY.

Its situated near Pauntley Court - home of Dick Whittington- looks like the school was supported by the landed gent at the Court.

The Education Act 1870 being one of the key pieces of legislation in school reform, so I guess this was a private rather than state school , but not in the sense of an Eton or Harrow :-)