Looking For Person Help Required

[jc26red]

If you obtain the birth cert, it might give you the address where she was born, maybe a home birth.  Or if you know the address, then look for neighbours. Maybe one will still be living that you can contact that might know the circumstances, they will probably have moved since then but may be still traceable.
Were any of your nan's sisters married? perhaps a living great uncle? or wider family, it must have been hard to keep that a secret.

How do you know your grandfather was mixed race? He may not have had any recent white heritage. 

[weste]

Perhaps try dna , use ancestrydna and then upload to other sites for more matches.  You possibly will get people related to his line, depends who has tested but you may need to put some work in and I would also test your mother as it will help to sort your matches out.

[aghadowey]

DNA might certainly help if another relative has also done a test (don't give up hoping for any kind of match as it can take a while for a match).
It may be that other relatives know more than you think (either remembering something at the time or told to them years ago or something an older relative mentioned in more recent times).
If your grandmother doesn't want anyone to know your grandfather's name it is difficult but would she talk about the War in general at all? Perhaps get her talking about what it was like when she was young, rationing of food and clothing, work, dances and movies, etc. You may not get a name but it would give you a clearer picture of her life during that period.