Somewhere warm and dry in February a Westfield's drive away from the West Midlands you say? Hmm, that's a big ask. How far are you expecting to drive?
From where you are I'd get myself kitted up with some proper winter clothing, pack light and try one of the following (drive times assume you live around Coventry):
Head to Fishguard (about 4hrs drive) and take the ferry to Rosslare in Ireland. Book a B&B or two for overnight stops and take advantage of the ROI's fabulous driving roads, stunning countryside and legendary welcoming people.
Or make the journey to Harwich (about 3hrs) and take the ferry to Esbjerg in Denmark. The trip to Copenhagen takes about 3hrs and is well worth the journey as it's one of the most cosmopolitan cities I've ever been to. Another beautiful country with fabulous people.
Finally, my choice would be to drive to Newcastle (about 3hrs 20) and board the ferry to Stavanger in Norway. Take a good digital camera and get loads of pictures of your car as you drive around the fjords nearby, which are unbelievably
stunning. (designed by Slartibardfast if you're a Douglas Adams fan, of course).
http://www.stavangertravel.co.uk/fjords-mountains-stavanger-norway.htm
I found the Norwegian people to be the second frendliest on Earth (after the Fijians) and loved every visit I made there. One word of caution though, if you like a drink, it's not exactly cheap...
Oh, and you'll need to check the ferry companies as they don't all do year round crossings.
Good luck and I hope you find a road trip you'll thoroughly enjoy.