In reply to JimboMac: Being someone who also jumped straight into alpine winter here are my thoughts:
1. It's cold, damn cold. Make sure you have a really decent pair of boots and soprt your glove system out. (Depending on the difficulty you may be best climbing in a good pair of light ski-touring boots.
2. You must be able to ski. Unless all you want to do is Alpine icefall climbing (which is not
really alpine winter) then you will be skiing into and out of routes (certainly in the mont blanc massif).
3. It's quiet. You're much more likely to be able to get on the routes you want.
4. Stable weather. You can only really climb with a good forecast, but when it's good it's often good for a few days in a row and there's no risk of afternoon thunder.
£2000 for a course in nothing compared to the costs of all the new equipment you'll have to buy.
Do you have any Scottish winter experience?
Having now done some summer alpine climbing it's definitely more chilled out, but I think I prefer winter for mixed/ice. Summer I mainly stick to rock.