There's a Joseph Hay, 1, with parents John and Mary, and his siblings, in Manchester in 1891, and he is with them in 1901, age 11.
I can't say who the nurse child in 1891 is.
Can we be sure that because your grandfather thought there wasn't a birth certificate to be found, that this is actually correct?
Anyway, if you change your mind, and fancy having another stab at finding a birth certificate for him, you can now get them digitally (and instantly) from the GRO for just £3.
John