No, the Family History Centres are part of the Church of Jesus Christ of Latter-Day Saints (Mormons), and are usually in rooms attached to one of their churches. It is part of their religion to trace their ancestors to baptise them into their church, and it is through the LDS that we have access to their huge holdings of filmed records stored in vaults in Salt Lake City, and also to many online records like the IGI. They are very generous and friendly in sharing these records with non-members, and no-one has ever mentioned their religion to me at their centres. Everyone there just has a common interest in genealogy. The filmed records are kept on spools and fed into the machine to read on a screen. Haven't you seen the machines for reading filmed records in public libraries? - These are just the same, and the volunteers there will show you how to use them. Like I wrote, look in the phone book under "Church of Jesus Christ of Latter-Day Saints" for the closest FHC.
If you are still in Christchurch there is one at 25 Fendalton Road - but don't know whether it is damaged. There are Mormon churches in most NZ cities - just ring to find out when they open the FHCs to non-members. There is a small fee to order in the films, but a lot of other resources are freely available in each centre.
In one of my earlier messages I gave you the film number to order the parish records for Saint Lukes, Baldwin, and Frances has said [below] that these haven't been included in the burial indexes - and that Charles Lewin paid for a plot so the family is probably there. Terrible story about poor Elizabeth.

Sue