'Cannon' has nothing to do with guns: it comes from candle-makers.
The candle-makers once lived around here, and the name first appears in 1190 as Candelwrichstrete, the street of the candle-wrights. Shortened again and again in local cockney speech, it settled as Cannon Street in the 17th century. The City's ward of Candlewick is still named after the street.