Catherine,
I fear that your pursuit of "bastardy records" will prove fruitless! I say this because I initially followed a similar line of inquiry in respect of the unknown father of my late mother (born in 1925) & her brother (b.1923).
However it soon became apparent (to me, at least) that they were born out of an ongoing & supportive relationship - which explained why my grandmother did not formally pursue my grandfather for financial support (nor did she seek any charitable assistance). Simply put, he was still married and thus unable to marry her - as I believe he would otherwise have done.
The spacing of the timings of the births of your mother & her sisters suggests to me a similar scenario and so, with this in mind, I feel that you might be better off redirecting your inquiry along the following lines.
You have said that you have the 3 birth certificates which means you already have a wealth of valuable information. Each certificate will contain the addresses of both the place of birth and the mother's usual place of residence at that time.
Are any of these addresses the same from certificate to certificate? Have you checked these addresses against the contemporary electoral registers?
Unless the address for the usual place of residence is an institution or a lodging house (or similar), although your grandmother might not feature on the relevant electoral registers you may still be able to extrapolate from them the name of the probable father.
Willyam