Vera is my grandma Edna’s sister on the GRO the only birthday I can find says she was born in
Sep 1916 and so does Freebmd. However :-
The 1939 census states her name as Vera Binns birthday written as 24 May 1916.
Her details are: Vera Binns nee Herbert born 1916 in Accrington (Registered in Haslingden)
Mothers name Letitia Herbert nee Johnson
Fathers name Fred Herbert
Her sister is Edna Herbert born Nov 1917
Am I missing something or did people change their birthdays to suit back then, just seems a bit odd to me.
Mind you my family has been nothing but strange 
Not sure what you found saying she was born in September 1916
It doesn't say that.
It says her Birth was Registered at the Registry Office in the September Quarter 1916
Which as stated that Quarter commenced on 1st July to 30 September 1916
After 1 January 1875 Parents had Only 42 days to Register a Birth or be Fined
If Vera was born 24th May
Her Mother did not Register the Birth before the April May June Quarter ended on 30th June 1916
But appears to have gone over
but on or before the 42nd day deadline on 2 July 1916
Which took the Registration into September Quarter 1916
1939 Register Births Transferred by the Enumerator can in my Experience can be often not believed correct.
Also a Birth Date on a Death Certificate may also not be Accurate as well.
Remember History is Not a Record of the Truth
It is a Record of What was "Said"
Which may Not be the Truth
You cannot Research the Past
Expecting 21st Century 100% Accuracy Eyes