For some reason, magazines print up their mailing labels way in advance, and it sounds like they'll add you to the next list they print. They probably also batch up their subscription updates, so it might be a couple of weeks until they add your name to the list.
If the magazine is a monthly, you get up to one month of delay if you happen to subscribe the day after the current issue comes out, then another month for the label nonsense, and the third month for the leisurely process of printing and mailing.
I'll be interested to hear if the weekly takes as long to start up, since their cycle is presumably a week rather than a month. Remember that Amazon say three months to everyone, for every magazine, so that's the worst case number.