Thank you so much everyone for responding so helpfully.
To answer the questions:
.. have you found the child in 1871? Who are they with?
I found the same record as Monica. I suspected this might be them but then was confused because Mary's surname was now Stevenson but I couldn't find a marriage record. I'd also missed the occupation matching so couldn't quite convince myself it was the same James, Mary & William (such common names, right?)
What was the name of the child born in 1867?
Where was the birth registered?
Who registered the birth?
What was the address?
It is necessary to know all the information on the birth reg.
Sorry, I should have included more on my OP. Images of William Stevenson's birth, marriage and death certs all attached this time.
...It may be worth obtaining some of the birth and death certs noted above to establish if they are the other children of the James Stevenson and Mary Gillespie you are looking for....
Thanks so much for that useful, I've obtained Richard Stevenson's birth certificate which looks very similar to William's (ran out of space to attach it to the post but parents names & occupations same as on William's and also "Illegitimate").
It's sounding like the consensus of opinion is that it's really likely that Mary Gillespie and James Stevenson just never married officially although they lived as if husband and wife throughout? I think I was thrown off by assuming the Mary's name would only change from Gillespie to Stevenson if there was a marriage record but I can see how naοve that assumption was now.
I think that unblocks me and gives me confidence that the Mary Stevenson/Gillespie & James Stevenson on the census records are my ancestors and let's me try and pin down some birth records for them (why did everyone choose such common names back then!!? :-))
Thank you so much everyone - it's really appreciated