Yes but having a gun like a pistol or shotgun for home defence would work better than an ice axe. I would only use my axe if he got close.
Society collapsed so the ex police military and other government with guns would use them. So if they decided that you had something they wanted your ice axe isn’t going to be much help.
But you have to take into account that the streets would be clogged. A 15 minute trip in a car could take hours even a day in an event like that if all utilities stopped working. Also the 9 million people living in the city would be trying to get out just like you.
How ever given your millitary background you would have a better chance at survival.