You can measure diameter and pitch with a caliper. It is easier though to measure pitch with a pitch gauge and they are cheap to buy. Between the two it will tell you if the the nut is metric or Imperial but chances are that it is metric. In metric threads are measured as threads per millimeter. In SAE they are measured in threads per inch.
That's only 1/2 your problem though because the collet has the be the right diameter and taper to fit into the forcing cone of the router shaft. It's highly likely that the company that made the router bought the collets and nuts from some other company and if you could find out who that is you might be able to get them there. It doesn't sound like that router was a very good deal if you are having that much trouble getting parts.
Someone I consider a master woodworker once told me that a master woodworker is not someone who never makes mistakes. He is someone who is able to cover them up so that no one can tell.