I live in a cold climate and find that it is the most important to keep your hands and ears warm! So it is worth getting some decent running gloves (I have good Craft ones with a mitten pullover which you can put on if it is really cold) and an ear warmer. Depending on how cold it is you will want long running tights and a long sleeved top but don't overdress as sweat will chill you if there is any kind of wind.
I had an ankle injury last year and found that aqua running and rowing didn't hurt it. Cycling was ok too but the cross trainer sometimes hurt. Also swimming might be ok but you have to see as kicking might irritate the ankle depending on your particular injury.
Last winter I ran numerous times in temperature between -10C and -17C (I live somewhere colder than UK) and didn't get a single cold; indeed I haven't had a cold in about 3 or 4 years as cycle outside every day in all weather and run in pretty much anything too. Gyms are definitely are more likely place to pick up infection - bugs love damp, warm places!
Don't overdress! I am a really cold person but you soon warm up once running and don't want to sweat too much as that is when you get chilled. An ear warmer is great and much better than a hat in general UK winter conditions. Watch out if you have shoes with big vents as they can be cold.
But as others say just be tough and get out the door as once out it is pretty unlikely you will go back.
Annajo - thanks for the link! There looks like a good route from Avenue Louise in Brussels which I am staying just off. Might try one of the Sydney routes when I am there too .
Split runs is a good idea too although it is often to get away in the evenings from the break off discussions. I too think a morning run is a great way to see somewhere when you are on business; my favourites so far have been Grenoble and Budapest.