As per the title. I'm thinking German style wooden huts etc.
Christmas markets are not really part of the British tradition in the way that they are elsewhere - at least not since Oliver Cromwell.
There are traditional German (and other) Christmas markets in various places in Britain, as shown in the above link. However none seem to be close to the Lake District.
It's not exactly the answer you were looking for, in that it's technically not the Lake District, and not really German, but it may be close enough on both counts to be useful to you.
Of sorts, Ulverston has a large Christmas market every year, called the Ulverston Dickensian Christmas Festival. It has grown quite large over the years, and takes in most of the central streets of the town. Quite a few locals (I'm one) dress up in Dickensian attire, and get into the spirit of things.
I think you could safely say the market has German elements to it, bratwurst vendors, traditional breads, cheeses, lots of local produce and art, fun rides for the children, and yes wooden huts. There's a good blog post. Ulverston is extremely close to Lake District, about 2 miles south, 15-20 minutes to Coniston.