If the School was in “Battle Hill Terrace”, the HLHS publication of “Vanishing Hexham Street and Place Names” says that this terrace was at “22 Battle Hill, top of south side”.
No 22 is no longer there, having been replaced by a building called Gibson House. However, if you look in the HLHS’s Photo Archive at the Hexham>Streets>Battle Hill Gallery (https://www.hexhamhistorian.org//historic-hexham/photograph-archive/photo-archive-choices/hexham-choices/streets-choices/battle-hill-gallery/) you can see early photos of that part of Battle Hill before the demolition and houses we think are numbered 20-24. There is an arch and alleyway between Nos 20 and 22, which could lead to “Battle Hill Terrace”, which was possible a row of buildings behind No 22. There is also a photo (CD0359) that shows the top end of Nos 22-24, with a possible alleyway running up besides No 24. Another possibility?
There is an online old map (1896 https://maps.nls.uk/view/132268799) that shows the street layout and at the top (W end) of Battle Hill you can see the alleyway abd buidings behind (i.e. to the S) of Battle Hill.