Will I really be able to drive my XK with 20" snow tires in the winter?

 

Absolutely with the correct tires. The XK vehicle is not unusual, it's another vehicle. The 20" also makes no difference.

Proper tire selection will be the essential part of winter driving. I would suggest reviewing the TireRack website to complete some research and ratings on winter thread tires. While there is no obligation to buy from TireRack, their website provides great comparisons of Winter and A/S tires.

It is a great site to start your research.

 

If you're in snow, you have a few things against you. 20" rims are far from the best choice due to the smaller footprint and the width make you ride more on top snow rather than biting through it. But as RichZak says, a good set of winter tires will make a huge difference.
Take a gander at the Conti DWS-06 for light snow conditions.

If your weather is harsh and long enough to be forced to consider something like Blizzacks, then just keep you cat indoors.

Vince

BTW: I Stopped buying from TireRack years ago. I go to my local dealer and they price match.... Keep you money local if you can!