Number 135 in the Roud Folk Song Index. Scottish and English forms are generally fairly distinct (though obviously related), with Scottish examples appearing earlier in print and beginning "O lassie art thou sleeping yet" (Burns wrote more than one song based on these) and English forms identifying the male protagonist as a soldier and bearing titles like Cold Blow and a Rainy Night or Cold Haily Rainy Night. English forms overlap with The Cottage in the Wood/Forty Long Miles (and other titles: Roud 608) but that is probably best considered a separate song, though sharing a fair few motifs. (Kennedy groups them together; see Ballad Index comments above).

There are a couple of 19th century broadside editions by Pitts at  http://www.bodley.ox.ac.uk/ballads/ballads.htm:

Cold haily rainy night

MacColl (see Ballad Index ref above) refers to a late 17th century broadside song, John's Earnest Request, but that's an earlier example of the same genre rather than an ancestor. There are two copies at the Bodleian:

John's earnest request: or, Betty's compassionate love extended to him in a time of distress