I'm assuming you mean 6 speed as there is (to the best of my knowledge anyway) no such thing as a 12 speed cassette.
The problem is that a 6 speed cogset is actually a freewheel mechanism with six cogs attached to it that screws onto a threaded hub whereas 8 speed and up cassettes generally slide onto a freehub body.
The only way it'll work is if the wheel you buy is of the old design which accepts screw on freewheels and without knowing what wheels you're looking at no one can guess if it will work.
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.
The wheels are "VINTAGE Raleigh 410 Sportif" and the cassette I plan on putting on them are whatever comes stock on the mid 80's Miyata bikes... I think maybe 110 or something close to bottom of that line.
Current 79 Nishiki Royal, Jeunet 620, 59 Crown Royal, notable previous bikes P.K. Ripper loop tail, Kawahara Laser Lite, Paramount Track full chrome, Raliegh Internatioanl, Motobecan Super Mirage.
In this case for vintage correct exceptable you can put most any 6/7 speed vintage strait, uniglide or modern Hyberglide. I would recomend a modernd Hyberglide type 7speed freewheel and a better quailty 8 speed chain.
You're doing a huge downgrade. I'm almost certain (and a quick check of the catalogs confirms) that even the low end Miyatas of the mid 80s had aluminum rims. And you want to replace them with steel. Do not do this.