Photographing Weddings in Winter
The major and most pressing concern that a photography must deal with at a winter wedding is light, and lack of it. Light and the quality of light has a major effect on photography and how it looks. There are a number of ways around it, but in general this is your major issue and the one to take greatest care about.
Light
When you are going to shoot a wedding in the winter season take time to find out when the sunset is and how this compares with the period immediately after the wedding ceremony, which is when a lot of the shots will be taken. In the UK, the sun can start setting earlier than 4pm, so earlier weddings give you more scope.
Actually all is not lost if you are due to take photographs just as the sun is setting, because although quite weak at that time of year, the best light for a soft diffused effect is often a couple of hours after the sun rises and before it sets as the light is just harsh. It is also worth noting that the light is very orange at this time of day too, and balance this against your subjects.
Additionally, you can use what light you have and use a reflector, either a bought one or a piece of white or shiny paper to be held to catch the light and drive it up to the face of your subjects. You can get ones in silver, white and gold for different light effects, just take care not to blind your subjects!
Artificial Light
Yes, absolutely, you can use flash, and drag the shutter in order to create an ambient light in places that are naturally dark, such as churches and reception halls, however, sometimes flash isn't allowed, so you would need to check. Lots of photographers use video lights on stands to highlight the bride and groom at important moments in the ceremony, first dance, etc and these can be put into effect if you are taking photographs indoors. If you have a battery pack for the light you can also use it to supplement the light there.
Studio Lighting
Studio lights can be transported with both battery packs for outside and use where there is no power, and also with power cord if you know you will be near a plug point. These can be really useful for getting group shots done on horrible grey days, as you can relax a little on time and take them indoors out of the cold and wet potentially.
Other issues you face surround making sure the bride and groom don't get cold, and my advice is to work quickly, get them in and out so they aren't continually cold, and if possible shoot them in front of a plate glass window so they get natural light, but also get to stay warm and comfortable, as they should look relaxed and happy whatever the weather throws at you.