Hi Ruth,
the information that you have been given is correct. Gladstone Terrace was indeed just off Adelaide Street which is off the A508 Barrack Road. The area is indeed known as Semilong but the is a number of schools that your mum could have attended.
As an infant, she may have attended St. Pauls School, which was situated in Semilong Road. If she did attend that school she would probably have stayed there through her "junior" years (until she was about 11 years old. she may have gone on then to attend St. Georges School along the Barrack Road.
In the 1950's this school was a secondary modern school.
Another alternative would be Spring Lane School as an infant and junior.
A 3rd choice would have been Military Road School, which (of this point I am not 100% certain) I believe was a secondary school (seniors).
Of the schools that I have mentioned, only Spring Lane is still open.
I attended St. Paul's from 1951 to 1957 and my sister attended St Paul's from 1947 to 1953 when she started at St Georges where she stayed until 1957.
Incidentally, there are a few books about "old Northampton" which feature a photograph of the residents of Gladstone Terrace at the time of the coronation of King George 6th (1937).
regards
Mick