Riders give up cycling in winter for three reasons: they get wet, they get cold, and they cannot see. Each has a straightforward fix, and none of them involves being especially hardy.
Wet is a fender problem, not a rain problem
On a road that is merely damp, almost all the water that reaches you comes off your own wheels. Full-length fenders with a long front mudflap eliminate perhaps ninety per cent of it. Stubby clip-on guards do not — the spray you care about comes off the bottom of the front wheel, well ahead of where a short guard reaches.
Fit proper fenders and a light shower becomes a non-event.
Cold is a hands and feet problem
Your core warms up within five minutes of riding and stays warm. Your extremities do not, because they are stationary in the airflow. Dress the torso lightly enough that you do not overheat, and spend the money on gloves and overshoes instead.
The classic mistake is a heavy coat and thin gloves, which produces someone who is sweating and whose fingers hurt.
Seeing and being seen
Winter commuting happens in darkness at both ends of the day. A front light with a shaped beam lights the road without dazzling oncoming riders; a wide-angle rear light matters even more, because it determines whether a driver approaching from an angle registers you at all.
Both should be rechargeable and both should live on the bicycle. A light you have to remember to bring is a light you will eventually forget.
Salt is the real enemy
Road salt corrodes far more aggressively than water. Once a week through the salted months, rinse the frame and drivetrain with plain water, dry the chain and re-oil it. Five minutes, and it is the difference between a drivetrain lasting one winter and five.
Tyres, briefly
Drop the pressure fifteen per cent from your summer setting for a bigger contact patch on wet surfaces. Unless you regularly meet actual ice, studded tyres are more trouble than they are worth in most cities.
The honest summary
Fenders, decent gloves, two rechargeable lights and a weekly rinse. That is essentially the whole of winter cycling, and it costs less than a month of public transport.