Anita I think I've cracked it - or at least in part. You are going to have to suspend disbelief though and bear with me as it's taken me hours and I'm still not sure what I've found
I think the GRO may have rejected your birth cert application because you specified her father as John Walster? Just a guess but I think her parents were not married and her father was John Green.
Anyway - here are some census details
1871 4522 137 15
8 St Thomas Row Leeds Bk Byron Street
John Green head 65 cylinder fitter Southwell
Olivia wife 50 Sutton
Walter son 21 moulder Mansfield
Olivia dau 17 Mansfield
Willoughby son 13 wire worker Barnsley
Thomas son 9 scholar Leeds
1861 RG9 3446 46 27
Barnsley
John Green head 55 grocer Notts Sutton
Olivia wife 44 Notts Sutton
Ann dau 20 Yorks Rotherham
John son 16 Notts Stamford
Walter son 9 Mansfield
Olivia dau 7 Mansfield
Willoughby son 3 Barnsley
Mary Jane dau 7 months Barnsley
Martha dau 7 months Barnsley
I cannot find a marriage for a John Green to an Olivia (or similar) anywhere in Notts or Yorkshire. However I find a couple of births
Walter Walster Mar 1852 Mansfield 7b 71
Martha and Mary Walster Sep 1860 Barnsley 9c 119
also a death that may be relevant although she doesn't appear on the census
Rebecca Green Walster Barnsley Sep 1856
What do you think? Heather