This doesn't come to a conclusion but here goes...
As you will know, in the 1939 Register, Claypit Farm is preceded by Bulwark Road, Fairfield Road, St Tecla Road and followed by St Tecla Farm, Hendersons Library, Bulwark Stores, Alpha Road
Looking at this map from 1922, the thing that stands out is that there is a Fairfield Farm next to St Tecla Farm which doesn't seems to be there in the 1939 Register, but is in the place where Claypit Farm might be expected.
http://tinyurl.com/3ewpj2wuConversely, in the 1921 census Fairfield Farm precedes St Tecla Farm. (Unfortunately Fairfield Farm is the first entry in this particular census book so there are no preceding entries, but subsequent entries move away to Burnt Barn and New House Farm.)
The simplest explanation for this is that Fairfield Farm was renamed to Claypit Farm between 1921 and 1939.
On this map from 1949/1954 Fairfield Farm has been replaced by a group of structures that may be postwar prefabs. But you say that the farm was in your family until 1963, so that doesn't fit with my theory.
http://tinyurl.com/2jvcc8kbOn the 1971 map the whole area is built up, but there are modern streets named Fairfield and St Tecla Roads just off Bulwark Road (all still there obviously). According to newspaper ads the address of the Severn Bridge Social Club was Bulwark Road.