There... did I miss anything?
I contact some people from back in high school and get into the rewarding world of smuggling. (As far as the government is concerned in such a senario, I'm already a criminal so I might as well play the role.) Much like happened with alcohol prohibition, people who were in the business of alcohol, who were otherwise law abiding people, got into the smuggling business to pay the bills. The police only catch the dumb smugglers. (Don't speed in a stolen car with a trunk full of Heroin down the interstate with no cover story and you will probably be fine.) Leaving the smart ones to go on with life.
The end result is that the "War on Guns" will be just as successful as the war on drugs. Instead of having to just worry about the people who misuse drugs/guns and hurt others, the police will have to go after: the new black market where the only way to settle disputes is with violence, people who own guns but aren't hurting anyone, police corruption as a result of the profits involved. End result, the problem becomes 4 times worse than it was, as can be seen with the War on drugs. (25% of violence a result of drug use, 75% of violence a result of the illegal drug trade.)
We already have 25% of the worlds prison population under the War on Drugs. We could likely see out prison population increase by another million or so. Meaning more prisons being build, since building school is for losers.
All the while, drugs and guns will be readily available, though at their vastly inflated black market price. Last I checked, Glocks in the UK are readily available, but go for about $5K right now. People who can't afford that have to make do with sawed off shotguns.