Just to clarify.
Bilston itself was, and is now, I believe, in the registration district of Wolverhampton , except for a period durng the 20th century when it formed it's own registration district.
However, the boundary with Sedgley is, I think, less than half a mile from the town centre on the Highfields side of the town, so it is perfectly possible for people who thought that they had been born in Bilston (and even lived there all their lives) to have been just outside the boundary and so registered in Sedgley (hence Dudley Registration District, and a different county).
It is therefore, a good idea to check both Wolverhampton and Dudley registration districts for people who say that they were born in Bilston in the 19th century. Wolverhampton Registry Office were very helpful when I applied for one certificate when the person of the same name registered in Wolverhampton turned out not to be the one that I was looking for and found the birth registered at Dudley, and after calling me, forwarded the application direct to Dudley.