I agree with Maiden Stone that confirmation is a very likely reason fo be baptised at that age.
If you think that grandmother and grandchild were together, then an address for Elizabeth might lead you to a local church record.
Continuing the confirmation theory.
The baptismal ceremony in 1947 may have been a conditional baptism, if there was uncertainty over whether the confirmation candidate had already been baptised, or if they couldn't provide proof of prior baptism.
Either of the above reasons may have been likely in this case because:
The child was living hundreds of miles away from where she was born.
She may have been living with her grandmother in 1947; grandma may not have known if, or where and when, she was baptised or how to find out.
Parent/s may have forgotten name of church.
The church where she was baptised when a baby may have been bombed.
Transcriptions of some recent C. of E. baptism registers in the wider Bury district are on LAN-OPC. Catholic baptism registers are subject to 100 year privacy rule (maybe 110 years now due to increased longevity) and so shouldn't be online. I don't know what privacy policies of Non-conformist denominations are.
