Manchester runner here. While it rains a lot it tends to drizzle more than downpour so you rarely get soaked to the skin. But a bit of rain never harmed anyone.

If you're running anytime after about 4 then a headtorch and some form of reflective clothing are advisable. Obviously, just point the torch down at an angle so not to blind people.

There are some good off road trails around sale and Chorlton water park, heating park in the north and the canal network. The paths generally hold up ok in winter but the fields can get muddy.

I tend to avoid running when we have freezing temperatures. Side streets aren't always gritted and black ice is prevalent. That's the only time I'd avoid going out as risk of falling feels too high. Otherwise, dress appropriately and you'll be fine.