Not sure if this is the greatest idea, tbh.
I started running at your age. Eighteen months later and I'm just about to run a half-marathon.
I think you might be pushing it. It is always better to have a couple of weeks' slack in a training plan, and your timescale is so tight that a week here and a week there might throw you off completely. You won't always want to run, even if you love it. You might need to be doing other things. You might (will) get injured/ill. I lost a few weeks going from 10K (and I could run that distance easily) to HM - a bad cold, toes so bruised I could barely walk.
Being overweight will put you at increased injury risk (esp. of stress fracture), as will the rapid increase in mileage.
Over-winter training brings its own challenges - are you happy to run in the dark? In gales and freezing rain/winds? You might lose a couple of weeks to snow/ice (unless you can run on a treadmill).
Are you going to be running alone? I'm pretty motivated but I would not have kept on running were it not for an informal network of local new runners - we often run together and encourage each other. Running can be really psychologically hard sometimes!
I would say absolutely start running, but take it more slowly, so that it becomes part of your life and not just a short-term project.