The last 2 winters on an 11 mile each way commute in London:
base layer, long sleeved jersey and a windproof
Altura night vision jacket. The three layers have always been enough except for one day last winter when I shoved a sweater on as well. The sweater came off after 2 miles. I have a rain jacket, but only use it if it's
really raining. Winter tights are great. 2 pairs of socks (woollen hiking socks over normal), and overshoes when very wet (I ride a brompton, which has a low bottom bracket = wetter feet). Winter gloves and occasionally liners when very cold.
Skull cap, yes. Keeping head and extremities warm is more important than anything.
My experience is that cheap stuff ( cheapo base layers from sports direct and Amazon, cheapo glove liners etc) works well, and stuff which dries quickly, such as lycra thermal tights, is better than waterproof, even in wet snow. But your mileage may vary.
Sometimes I say f*** *t and throw the bike on the train (this is why I ride a Brompton - well, one of the reasons, anyway). On such days my ride is only about 2.5 miles each way and I just wear normal clothes.
I look at teh collection of cycle clothing I have and somethimes wonder at it...but it all gets used. A lot.