You're in the NE, it's not that cold. Just went out this morning without a coat (cozy jumper though). And I'm assuming you're not that rural or your kids would be on a school bus so don't think you need particularly technical clothing.
Look at a few blogs and see what you think. Midlifechic is in the NW and is a very classic dresser and has talked about dressing for 'not London' so might be worth looking at what style of clothes she wears (her outfits of the day are good for seeing how she dresses for e.g. dog walking).
I just wear whatever I wore to work, I sometimes change shoes if I was wearing something particularly flimsy that wouldn't cope with the mile to school and back. And maybe add a jumper if I've just had on a silk blouse at work. If I WFH I only change if I've worn my pyjama bottoms all day (I don't sleep in them but they are super cozy).
Brands I think are good include Hush, e.g. I have a lovely felted wool dress from there that is super warm (not this one but similar, I wear it with woollen tights and ankle boots with a leather jacket and big scarf. Or their harems are lovely and comfy, I wear mine with white trainers, cashmere jumper and whatever coat I fancy (I have mainly wool coats but also my leather jacket and an old Boden tweed jacket for winter. I do have the dreaded Joules raincoat but that is reserved for the 2 or 3 days a year when the heavens open and I go fully practical with my green noras and an umberella).
What coats do you have? If you have e.g. a parka, a leather jacket, a classic styled wool coat and then one properly waterproof coat that is thin enough to wear in the summer but has room for a warm jumper in the winter then you're sorted for any occasion. Then trouser shape and style of your choice in blue, black, grey, khaki. Shoes: one sensible pair of heavy duty walking shoes, ankle boots in your preferred colour, pair of white leather trainers (M&S do excellent dupes of some of the more designer brands), maybe a pair of knee high boots as well in your preferred colour. Maybe a few wool or cord dresses and woollen tights (much warmer than jeans)? And then whatever tshirt and jumpers you like in whatever colours you like. Don't think 'I need to dress country', I come from the far north of Scotland (NE of England is practically the equator for me) and at home you see everything from those who dress like they did 30 years ago to those who are super stylish or eccentric. There is nowhere in the UK where the weather is so bad that you have to ignore fashion and purely dress practically for doing something as mundane as the school run. Presumably your DC will be in uniform?