If there's no headstone or the birth date is not recorded on it, should there be one, there's a slim chance the birth date may have been recorded at her baptism. If she was baptised at St Barnabas, the records are held at the Cheshire Records Office. I think they would charge for a search, perhaps if you email them first to find out.
Your sister was registered, Barbara E Graham, 1940 Wirral, Vol 8a Page 1277, it may work out cheaper to order the birth cert from the GRO, it will definitely have the required information on that. Good Luck.
God help us,
Don't send Jesus,
This is no place for children.