These kind of searches are often guess work May, just picking up on possible clues to try and trace people. I think from what you have said, if a letter was written by Annie when she handed over Mary to the Wood family, I would guess that they probably were not family and they wanted something in writing to confirm the handing over of Mary to them (all of this well before the start of the legal adoption process and laws in the 20th Century). I would think that the Woods family could either be friends or connected to them in some way. Other possibilities are that some charitable group/church may have arranged the adoption.
However, with my optimisitc hat on

There is an interesting possibility in the censuses. Going by the address on Mary's birth entry, there is this possible Annie. Two things stand out: the lodger in 1881 and the address for the 1891 census entry which seems very close:
1881 - all the Whitelaw family born in Old Monkland:
Elizabeth Whitelaw 44
Annie J Whitelaw 19,
John Whitelaw 17
Elizabeth J Whitelaw 12
James J B Whitelaw 10
Joseph Wood 31, lodger, engine fitter born England
Wm B Williamson 53, lodger, labourer, b. Kirkcaldy, Fifeshire
Address: 121 Springburn Rd, Glasgow Maryhill
1891:
Elizabeth Whitelaw 54 b. Langloan, Lanarkshire
Annie Whitelaw 29, b. Boot Machinist, b. Coatbridge Lanarkshire
John Whitelaw 27, railway clerk b. Glasgow
Elizabeth Whitelaw 22, Boot Top Fitter b. Glasgow
James J B Whitelaw 20, Coal Masters Bookkeeper b. Glasgow
Address: 111 Alexandra Parade, Dennistoun, Glasgow
What was the occupation of Annie on Mary's birth cert?
Monica