Well, I have at least found the marriage reference:
Mary E. Lane married a Mr. White-Harris in the March quarter of 1967 (that is, the marriage was registered in that quarter). The registration district was the New Forest, Volume 6B, Page 1005.
The marriage certificate might reveal your mother's father's name (it usually does, though not always if the father was deceased or the person was illegitimate, for example). If the father's name is not revealed, that might be negative evidence for Mary E. Lane being illegitimate, and point towards that birth reference in Bristol in 1947.