The definition of "nip" is to pinch or to steal. Apparently, Nippers has a couple of pastimes that have to do with taking a pinch of something and possibly dealing with people who take a pinch of something else. An established informal usage of "nip" is to mean to take a small amount of alcohol to drink, which Nippers does at lunch thereby steadying his nerves and his focus, which are apparently unsteadied by more than nipping t alcohol during the night. The "clients" who come to see him who look like "bill collectors" could be people he places gambling bets with thereby trying to nip a little good luck on the side. Based on these two circumstances relating to the meaning of "nip," Nippers is very well suited to his nickname.