I believe Bingley has it right. There was a movement in England, probably starting in the late C19, possibly a bit earlier, that promoted healthy living. Their regimen included not over-eating or drinking too much (or abstaining altogether from alcohol, for extremists) wearing loose clothing that allowed air to circulate, and taking cold baths each morning, whatever the ambient temperature. If these cold baths could be taken in the river, lake or sea, so much the better, but a plunge into the bath tub at home was a good substitute. It was particularly recommended for young men who were suffering from the less morally respectable symptoms associated with the onset of puberty. (The time when a young man says good-bye to his boyhood and looks forward to his adultery) thanks for that one, Jackie!!

The bath was commonly referred to as, "the tub", so "cold-tubbing" becomes immediately comprehensible (in meaning, if not in practice!)
The "cold-tubbing races" are, I presume, the morally-superior Anglo-Saxons, as compared to the degenerate Latin races. (Whoops - time I came back into the C21)