OK, going from nothing to a Half marathon is a quite something.
I did it in about 18months, in my 50s. If you are younger, then a year may well be possible (remember, if you are not fit enough and injury free, you can defer a place to the following year)
Running won't make you lose weight (indeed once you are running a lot, you'll be ravenous) so you need to look at that separately, and find a way to eat less but healthily.
Start running with C25K. That's a 9 week programme, but might take a week or few longer if you find any part too tough or are ill/injured. Various apps will steer you through it, and Zombies!Run! has released a training app which is similar running training but with added undead incentive.
Start parkrunning just before completing C25K (as you don't need to run the whole thing) and enjoy this distance for a while. Maybe see if there's a running group or a branch of the Good Gym near you.
Then take another 3 months to double your 5k to 10k. This is actually easier than going from 0 - 5K! Enter some 10k events.
Then find a Half training plan. If you google, you will find lots. But if it helps you to have a recommendation, thus is one I found worked really well (led to my Personal Best)
raceforlife.cancerresearchuk.org/sites/default/files/RFL%20Half%20Marathon%20Intermediate%20Training%20Plan.pdf
And have fun!
I was a fat non-runner 3 years ago. Who would have laughed in your face if someone suggested running. I was a somewhat thinner brand new runner 2 years ago. Now, I run Half marathons because I enjoy them (I'm halfway to me aim of doing 8 this year)
You can do this!
But I must sound a note of caution - a rapid training plan risks injury. If things start to hurt, slow down and defer your place to 2021