Steve,
With reference to your enquiry:
"Has anyone come across/heard of an example, where a child is born out of wedlock, the fathers name is missing off the birth certificate, the child takes the mothers maiden name, at a later date the parents marry and the father "adopts" their own child, and the child then takes the father's name."
Hopefully this may help:
When my older brother was born in September 1945 his (our) parents were not married and so his given birth surname was that of our mother.
However, after they married in April 1947, one of the first things that they did was to go to their local juvenile court (in Shrewsbury) to jointly formally adopt their first child. Thereafter he carried his father’s surname.
Although both he & I were not made aware of this by our mother until shortly after our father's death, I still took a certain vicarious pleasure in pointing out to him that he had appeared in court at such an early age.
By the way, this is not anecdotal as I still retain a photocopy of the relevant court document - having passed the original on to my brother.
Willyam