The origin of sprog is something of a mystery. It apparently started out as derogatory slang in the British armed services during World War II, first appearing in print in 1941. Sprog didn't originally refer to a child. A sprog was a new recruit or trainee, an inferior in both rank and social status. As soldiers and sailors left the services at the end of the war, we find sprog showing up in civilian use around 1945, for the first time being used to mean a child or baby.
I can verify shaneystar2's answer, I served in the army for ten years, and new recruits were always known as sprogs, and my nephew, now serving still calls them sprogs.