I understand what it's like in country towns. It's all a bit 'car seats optional' and I've even seen kids rock up to school in wagon backs. So I can understand how frustrating it is feeling like a driving teetotaller!
Do you have a pram with a rain cover? When I walked (a lot!) with DD in Canada's winter (down to minus 20) the rain cover was really important in keeping her from breathing in too much icy air. It was easier to rug her up in the pram too as I could line it with a blanket then wrap a blanket over her legs and feet as well.
As an extra incentive for you, to help you get up 30 minutes early, remember that you could potentially be charged with child endangerment in a scenario like this (not sure on the laws exactly though), which would mean bye bye teaching career. Even if you weren't charged with anything, it could still do lasting damage to your reputation as someone responsible for children.