Hi Greendoor,
Do you have the censuses on your Grandmother? If so, what did they say about her birthplace and whom she was living with?
Perhaps that will clue you in on her parentage.
Regarding two different names for her father, do you mean different forenames or full names? I think she may do not know her father's name because of the illegitimacy?
Have you tried to see the list of all Doherty births for a particular county or particular town seeing any similar name to your Grandmother's within the range of years?
Also, you would have to consider about the spelling of the surname Doherty--it may be mistranscribed or incorrectly spelt.
Go back to 1891 and 1901 Censuses will help you to determine where to look for her birth entry.
Best wishes,
Tees