In short: No, no, no.
You can technically do this, unluckily, but it is highly inadvisable. It is a pity that it just looks like one could do it (1/2'' is not far off compared to 12mm, and 1/4'' is even closer to 6mm) and that with some luck, it indeed seems to "work just fine".
That is true for the other way around, too. You can hardly tell a difference with the naked eye, and the bits will fit in "just fine" either way.
I've seen people do it, too, and they so far got away without accident or even physical injury. (Which makes the whole matter worse because that "proves" that it works.)
However, using an imperial bit in the corresponding (well, almost corresponding) metric collet is dangerous both during the operation and when using the collet with your 6mm bits as intended afterwards.
What happens is that you irreversibly widen your collet by a tiny amount (too small to see, but certainly enough to be on the dangerous side). If you are unlucky, the whole thing blows to smithereens at 25k RPM with shrapnel flying all over the place and the bit come off flying in a random direction.
If you are lucky it "works fine", but now the collet no longer grips the original bits firmly and safely, so you have the risk of a bit flying away in a random direction every time you use the tool "properly" with the correct bits. Plus, you never know if the collet isn't going to break a month or two later, anyway.
The same is true for the opposite case, using a metric bit in an imperial collet. Only in this case, you force the clamps together much more than originally intended. There is a bit of tolerance, so you can certainly fasten the screw harshly enough to do those extra 0.3 millimeters. But it's not what the tool manufacturer intended you to do, or planned in.
The bit isn't going to have a sufficiently firm, safe grip, and you cannot be sure whether you have structurally damaged the collet by bending the little clamps too harshly.