Much of the following information comes from the very helpful Gill Clegg's Chiswick History page.
1707 a charity school was founded at St Nicholas church near the river. It expanded until in 1813 the boys were rehoused at Turnham Green and by 1819 this was a National School for boys.
The school seems to have had two endings: in 1848 it moved to a new school building in Essex Place - north of Turnham Green and demolished c.1968;. and in 1905 it was replaced by Belmont School in Belmont Road - still standing, north of Turnham Green. Perhaps the Essex Place was an extension rather than a replacement, or the school moved first to Essex Place and then to the nearby Belmont building.
Either way, we have no definitive explanation for the plaque's reference to "old building" from which it was removed in 1880.
Background: in 1811 the National Society for the Education of the Poor in the Principles of the Established Church was formed and gradually took over the charity schools, renaming them as National Schools.
From Brentford & Chiswick Local History Society : Prior to the Vestry Hall being completed in 1876 the Chiswick Vestry met in the “Boys’ National Schools at Turnham Green”. The Vestry Hall was later developed into the Chiswick Town Hall which is right next door to the building with the plaque.
From Brentford & Chiswick Local History Society: a man born in 1900, talks about his school days 1909-12: “Close to Chiswick Town Hall there was an old National School building which was used as a handicraft centre for boys and a domestic science centre for girls.”