to keep quiet about something; my lips are sealed.
mum means mmmmmm (the sound made when putting your lips together)
came from Shakespeare quote:
"Seal up your lips and give no words but mum."
person 1: "Please don't tell anyone about my affair."
A criticism pertaining the management, when criticising middle management or other useless and expensive building hold backs, Gölök Buday, the Black Jester would say "MUMs the word," going back to the 2000 election when Maharishi's university of Management started getting noticed, ala John Hageline.
Products would cost less if they spent less of tedious corporate bureaucracy, MUMs the Word!