Can you plant grass seed in winter?
Yes, of course you can but it won't work how you'd imagine.
Sowing grass seed in winter is called "dormant seeding" the seeds you plant will lie in the ground, staying dormant until the Spring comes and warms up the soil.
This carries some risks:

Will it grow?
Your success or failure depends entirely on the weather / temperatures. If the soil is warm enough your grass seed will grow in the winter although it may not survive.
Another overlooked factor is the type of grass seed you use. Most popular varieties require a soil temperature of 9-12 degrees, you can get some fast growing grass seeds that can germinate in soil with temperatures as low as 5 degrees.
I wish I could give you a definitive answer, you can certainly grow grass seed in winter but it may not survive, it is often best to do the preparation in winter then wait until Spring or Autumn.
TOP TIP: Find out when is the best time to plant grass seed.

How to repair winter lawn damage?
British winters will wreak havoc on even the most cared for lawn. A unique mix of torrential rain, freezing temperatures, snow, frost and the occasional warm spell can leave your lawn looking like a right mess come the Spring.
One of the best ways to repair winter lawn damage is to prevent it happening in the first place! Just because it's cold and your grass is growing more slowly doesn't mean your job is done.

How to sow grass seed in winter
If you are completely determined to plant your grass seed this winter then here's how you should go about it for the best chance of success:
Winter is definitely not the best time to grow grass seed (especially in the shade) but doing something is better than doing nothing. Put the effort in during the winter and your lawn will Spring to life when the temperatures rise giving you the lush, healthy green lawn you dreamed of.