I found an article about false etymology on the Internet, which mentions this false explanation of "wake":
"Lead cups were used to drink ale or whiskey," proclaims the Internet message. "The combination would sometimes knock people out for a couple of days. Someone walking along the road would them for dead and prepare them for burial. They were laid out on the kitchen table for a couple of days and the family would gather around and eat and drink and wait and see if they would wake up. Hence the custom of holding a 'wake.'
The author, Richard Lederer, says that the correct meaning is this: "'Wake descends from the Middle English 'wakien,' 'to be awake,' and is cognate with the Latin 'vigil.' 'Wake' simply means, traditionally at least, that someone stays awake all night at the side of the casket on the night before the funeral."
A dictionary says:
an annual English parish festival formerly held in commemoration of the church's patron saint...the festivities originally connected with the wake of an English parish church...a watch held over the body of a dead person prior to burial and sometimes accompanied by festivity
I don't know if the following explanation is true, but supposedly it's part of a 19th century Scottish custom:
For several days the body was "Waked" - Members of the family, numbering 2 to 10 people, usually the young and unmarried, would watch over the body around-the-clock., to keep the spirit from falling to the Devil. Curtains or blinds were drawn until after the funeral.
Family and friends of the deceased would come and pay their last respects. Readings were made from the Bible, along with the singing of hymns, and conversing in low hushed tones. Neighbors would help by bringing extra chairs for the watchers or extra peat to help heat the house throughout the "Dead Days."
I guess if I were a historian or had access to some academic types who specialize in British history or etymology, I'd ask them, but this is the best I can do. I wonder what the real story is, if there is one.