Uppies and Downies is a game played in Workington, Cumbria, England. Basically, grown men chase a small ball through the town from one end to the other, upper town versus lower town. It sometimes gets quite violent.
"Ere marra, how did ya lose ya finger?"
"Uppies and downies pal, uppies and downies."
When a person drinks only one or two beers, then stops drinking, the person will inevitably become tired/drowsie. This phrase describes that phenomenon.
Since I've had lunch I've got the one beerdrowsies, should have drank three or four, at least then I'd be able to stay awake.