I tried the National Schools Admission records in the hope that they would give a d.o.b.(to give a better search in the 1939 register) and there might be an indication of what she was going to do as an occupation and the date she left school, which could get you a baby step forward.
The school her brother, James Vernon (d.o.b. 16 Jan 1901) attended from 24 Aug 1908 to 23 Dec 1914 was Helmington Row Board Mixed School which became Helmington Row Council Mixed School. The records I have access to only cover admissions up to early Jan 1910 which is too early for Bessie.
If you are in the Durham area, the Records Office Catalogue show they have the next book (16 Feb 1910 to 18 Jul 1927) their catalogue ref is E/W/F27. The earlier book had an alpha section at the front for surnames and also gave their entry number in the main admissions section - so easier to find her if she is in there.
Oddly, James V's school record shows he was going to be a baker when he left school, but he must have re-thought that as his RAF service record shows that when he joined in 1921 he had been working as a joiner.
Boo
