We can find very little about this school, it's all just hints and pointers.
AIM25's holdings for All Saints Church include some records about the parish school, presumably this one.
The earliest map we can find is from 1870 and shows the site with the plaques labelled "National School (boys and girls)". The OS maps of 1895 and 1913 show the same building labelled "Sunday School".
The National Archives hold Haggerston Road School logbooks 1879-1932. A guide from the London Metropolitan Archive (pdf) refers to a holding of Haggerston Road School, Lee Street, Admission and discharge register for boys, 1895 - 1931. Which suggests that the school also occupied a site on neighbouring Lee Street.
While the IWM says the school was demolished in the 1930s, the plaque says it was destroyed by bombs in 1941.
The hall now (2022) on the site was erected in 1955.
Do not confuse this school with Haggerston School, Weymouth Terrace. E2, designed in 1962 by Erno Goldfinger.