Things are moving forward. Kinda...
I applied for a copy of my father's birth certificate, as well as his and my mother's marriage certificate, with the GRO back on May 5th.
Today I received two copies of my father's birth certificate, but have not yet received their marriage certificate. I am a bit surprised that they weren't in the same envelope, as they were both part of one order. But hopefully I receive it in the next few days, so I can go ahead and apply for my UK Passport.
I was also a bit surprised that the birth certificate appeared to have images from an original hand entry of some sort on it, instead of retyped or a computerized printout of the information. The handwriting was a bit hard to decipher, and I still am not sure about all of it.
From what I can gather, my father was born at 26 Poplar Grove, which I believe is/was Stepping Hill Hospital in Manchester, and his father lived or worked at 2 Dundonald Street. I really can't make out the information about my grandfather's employment, even zooming in on the picture.
Oh well, it is more than I had known about.