Hello Emily, and thank you for your reply!
I suspect we're talking about different people, but you never know. Did your Emily Burlington emigrate to Canada at some point? I haven't dug into this research for a good year or two so I'm a bit hazy now, but I have a feeling the Emily who married a Watson was in Canada at some point.
I have a copy of a birth certificate for an Emily Burlington born 1901 Holytown to Joseph Burlington and Helen Burlington née Pickett, married 1894 Derry, Ireland. Is this your grandmother? I can't tell you anything about the adoption, sadly, but if you're sure about her birth name, year and place, then that's probably her, and so those would be the names of your great-grandparents.
The Emily/Emma Burlington I am looking for would (possibly) have been adopted by William James Burlington and Elizabeth (Lizzie) Burlington née Brown. They married in 1895 in Ballymena, Ireland. I have never been able to find a record of birth for her, either in Scotland or Ireland, and so I think there's a good chance she was adopted, or possibly even a foundling of some sort. It's impossible to know for sure, though, and I've never been able to trace any of Emily's descendents. She married twice that I know of, and although I was able to find a daughter born in 1942, she has since passed away.
I'd need to look back through my research, but I know that the Burlingtons were a large family and several of their children (who were therefore cousins) used many of the same names, and so I have a feeling that your grandmother and the Emily I am looking for were first cousins, as William and Joseph Burlington were brothers. Off the top of my head there was another brother, John, who appears with William and his family on the 1901 Glasgow census.
A lot of these people came over from Ireland around the turn of the century, and a lot of these families inter-married with one another. If I can help you in any way then of course I will, but aside from remembering something about Canada I'm afraid I don't have any details for you, other than I do recall seeing your Emily on a 1901 or 1911 census with some other Burlingtons, as I came across her once or twice whilst searching for the other Emily. However, if you choose to delve into this then be careful, as you don't want to confuse her with her cousin of the same name who appears on census records alongside William & Lizzle Burlington. They were around the same age and appear to have lived in the same area of Glasgow, but I do not believe they were the same girl.
If you ever happen to stumble across a record of birth for an Emily Burlington with parents' names William Joseph and Elizabeth then please let me know, as so far I'm drawing a blank.
How far have you managed to get with your research? I assume you have your grandmother's marriage certificate, but that it lists her adopted parents?