There are (or were) 3 major cemeteries / graveyards in Nantwich. If someone is in the burial register of St. Mary's, Nantwich, then before 1849, they would have been buried in the graveyard immediately adjacent to the church. As a result of the cholera epidemic of 1849, which claimed some 180 lives, the original graveyard began to run out of space and a new site out at The Barony, originally known as Finger Post Field, was purchased from the Workhouse. The first burial took place there on 21 July 1849. This became known as All Saints. (Somewhere in Rootschat for Cheshire you will find lots of images for it). The chapel of All Saints was opened in 1884, but was closed and demolished in 1985.
The 3rd was the municipal cemetery, which is separate again.
FamilySearch has indexed all the burial parish registers for Cheshire (errors and omissions excepted).
I am never clear whether an entry appears in a church's burial register if someone has a funeral there but is buried in a municipal cemetery. I think the answer is "Not usually, but..."
NB - in the UK "graveyard" is used for plots adjacent to a church, "cemetery" is an area not linked to a church, usually owned by a municipal corporation. All Saints' name is therefore a puzzle as it had no church to start with, then had one, and now doesn't have one. Cemetery or graveyard? Ah - the delights of the English language.
This link may be useful for the cemetery at Middlewich:
http://www.mikewalton.org.uk/mhsweb/mhslook_up.htm