Bits and collets in Milimeters


I have a question about bits and collets in metric system.

I have bought the x Carve and came with 1/4 collet and M15 nut for 1/4 HSS bit. I am living in Switzerland and I bought from here a 6 mm diameter bit. Unfortunately the bit doesn`t match with the collet.

I found one collet here that I would like to buy but the dimensions are different and I don`t know if will match with nut M15 that I have.

Dimensions of 1/4 collet
Diameter 12mm and 22mm long

Dimensions of 6mm collet
Diameter 11.5mm and 18mm long.

Would it be a problem with the nut M15?

Thanks for your help.