Hi
The burial guide (first page) gives the link to the Surrey deposited parish registers held by the Surrey History Centre. The list gives the dates held for baptism, marriages and burials for churches in Surrey. So for instance the deposited registers for burials at St Martins extends up to 1913 whereas the marriages go to 1996 and the baptisms 1993, which gives some indication of when St Martin's churchyard closed. Other than the asylum (Horton) cemetery and the Ashley Road cemetery these seemed to be the only burial areas in the town of Epsom. The other church registers have no burial registers.
Epsom was a civil registration district so a birth and death registration for Epsom covered a wider area than just the town of Epsom. The guide at the beginning gives the links to the information on Surrey registration districts. If you click on Epsom registration district you obtain information on all the areas covered..
For Epsom up to 1934 it was all these places
Ashtead, Banstead, Carshalton, Cheam, Chessington, Cobham, Cuddington, Epsom, Ewell, Fetcham, Great Bookham, Headley, Leatherhead, Little Bookham, Stoke D'Abernon and Sutton.
Do you know for definite the mother was living in Epsom in 1930? A burial, if found would give an address but no information on the parents (unless the child was buried in a purchased grave in which case others buried with her would be connected in some way). Illegitimate children were less likely to be baptised particularly if they died young so there may not be a baptised. However again you would need to know the whereabouts of the mother/parents to consider which churches were local and whose registers (if deposited) it might be possible to search.
Regards
Valda
