This is something that I too have been trying to follow up. Anywhere, locally, the american signals are bad news - why then is it that the fisheries authorities I have had any form of response from seem to be very protective of these alien invaders. "Unfortunately, we do not currently permit the trapping of crayfish...." I bet the local water bailifs and wardens don't have any problems "doing a random sample" knocked back with a nice light white wine on a Sunday afternoon. All in the interests of monitoring Signal Crayfish of course.
As to a licence - maybe all we need is the one from the DVLA so we can drive to crayfish hotspots!
For all you budding trappers, try this site.
Jarnn Fishing supply crayfish traps throughout America, Europe and the UK
You can get 2 traps, including postage to UK address for a little over a tenner!