Hello,
I attended Sandbach School in the pre-comprehensive days of the 11 plus examination. I do not recall ever hearing the title National School applied to Sandbach School. The school was a grammar school but different from state schools in that the local authority paid for your attendance if you passed the 11 plus (I recall the term direct grant school). There were old photos of sheep being raised in the grounds and boarders until just after the war (?). The junior school on the same site was private in my days (1970s).
My sister still lives in Sandbach and I will ask her if an old school building (once was scout hall) is still standing as this may have an inscription, a long shot but still worth investigating.
DrD.