Your questions are a little confusing, at least to me. I'm guessing you mean it's a 12 speed bike with a 6 speed "cassette" which in fact is probably a freewheel.
My best guess is the answer is yes, you can use the freewheel on the new wheels, and no, it doesn't matter that the wheels and bike are different brands. I'm not sure about the first part but the second part is rock solid. The fact that they are 27" could potentially be an issue. Is that what size the old wheels are?
Also, just because you can probably re-use the freewheel doesn't mean you should. If it's old, this might be a good time to replace it and install a new chain.
http://sheldonbrown.com/free-k7.html