Shirley I think, if you mean the square building in the centre of your right hand picture, then this would have been too far up to have been in Ann St, which ended at Newhall st. In fact I think it would be the Blue coat school , the position of which is shown on post 7 of
https://forum.birminghamhistory.co.uk/showthread.php?t=9516. Anne St school does appear, though not under that name.The 1845 and 1849 PO directories list in Ann St , in about the right position:
47 British & Foreign School, Miss Mary Ann Harris, mistress
48 Infant School,Mrs. Sarah Smith, mistress.
Either could be the school in question.
In 1855 it has become:
47 Temperance Hall,Benjamin Blackbam, secretary
48 InfantSchool,Mrs.SarahSmitb,mistress
The infant school is still there in 1862, but has gone by 1868. When the street was incorporated into Colmore Row 48 Ann St became 118 Colmore row and 47Ann St became 120 Colmore Row
Hope that is helpful
mike