Couch to 5k is a great program. I personally detest running on treadmills but I completely understand the need to be inside when it's unpleasant outside. I would suggest you get outside whenever you can - most people find running outside harder than on a treadmill, so the sooner you can get outside, the better! I would maybe do the first few weeks on the treadmill, until the weather is more conducive to being outside, and then when you're ready to go outside, maybe repeat the week you just did on the treadmill. That way, you've done the plan on the treadmill, so the only thing you're changing is the surroundings. If you opt to do the entire program on the treadmill then repeat it outside, I doubt very much you'd need to start at week one - more likely around week five or six, if even that.
Good luck - and remember to enjoy yourself!