Without evidence to the contrary, I'd assume his father was Thomas if that was written on his marriage cert. What was his occupation?

There are several potential problems with finding his birth cert in the early years of registration. Many births were not registered because there was no penalty for not doing so until 1875.
I would check all the Birmingham parish registers for a William with father Thomas Cox, from 1837 onwards. He may have been registered only as William, and the Henry added later, possibly at his baptism. And then check whether any possibles found had their births registered.

There are 15 William Cox births on FreeBMD registered in Birmingham up until 1845, with 2 William Henry.
Did you buy this one, which seems to be the most likely IF he was registered including Henry?
Births Dec 1839
Cox William Henry Birmingham 16 298

If that one was wrong you could ask the local County Record Office (is/was in Warwick) to check their records for a William (+ possibly Henry) birth cert 1837-1843 with father Thomas, which would cost a lot less than buying the 15 certs from the GRO. Or have you already approached the CRO?

Sue