Rory, this might sound a bit weird, but my first thought when I looked at the photo (before reading what you said about it), was that the lady looked Scottish. Now I know there is no such thing as looking Scottish, but this was just my first impression.

So, to the facts ...
I take it you are trying to find your grandmother's birth? What is her name?
Who is her mother?
What is the name of the grandmother in the photo? Was her husband still alive at the time this photo was taken? Or might she be wearing widow's weeds? I am wondering if the scarf tied at the neck means something? (if she is widowed)
I'm sure others will have other suggestions, but this looks like a studio portrait. Is there anything printed or written on the back? Any embossed words on the front? There were often clues in the photos, but the only thing I can see is the open window, and I'm not sure that that would have a particular meaning as it is a photographer's backdrop.
If your grandmother was illegitimate and this photo shows her with her grandmother there could have been any number of reasons that it was taken:
- perhaps a special event (her grandmother's birthday for example)
- the baby may have been adopted hence the last chance for a family photo (there may have been a name change which may explain why you cannot locate the birth?)
- the baby may have gone to live with the grandmother
or- they just wanted a nice family photo

Although illegitimate, she was obviously accepted by the family.
Not very helpful, but I'm not sure that you will be able to deduce much from the photo to help you in your quest.