It is actually embarrassing to post this.....
Direct from HM Busybodies :
You must not trap non-native crayfish without written consent from the Environment Agency (EA) in England or Natural Resources Wales in Wales.
Application to trap crayfish (PDF, 301KB, 2 pages)
You also need permission from the landowner and any relevant angling clubs.
If you trap crayfish without written permission you could be prosecuted.
If your application is approved you’ll get:
permission to trap using approved traps, see the application form for details
identity tags for each trap, these must be attached to traps
a catch return form, use this to keep a record of crayfish you catch
It takes 10 to 20 days to process your application. It may take longer if the water has a conservation designation such as a Site of Special Scientific Interest (SSSI).
In some areas you also need a licence to keep non-native crayfish alive after trapping.
Apply for a licence to keep non-native crayfish
If you are a fish farm owner or fish farm employee you don’t need consent to trap non-native crayfish but you must comply with trap rules.
Native crayfish
Native crayfish are a protected species – you can only trap them for scientific purposes and not to eat or sell.
To do this you need to register with Natural England and follow instructions on the class licence for white clawed crayfish.
In Wales contact Natural Resources Wales.
Read more about about your problem with invasive foreign crayfish species threatening the extinction of the native white clawed crayfish.
Six different foreign crayfish, some of which are crayfish plague carriers to which the native species is susceptible. Crikey!
Wasn’t aware that the foreign species also threatened erosion of river banks with deep burrowing.
Good capsule summary at:
https://www.buglife.org.uk/crayfish
Didn’t know my innocent question based on gustatory memories would lead down this path.
My guess as to the prohibition on trapping the non-native varieties is to prevent further spread of the plague... one would think that a more sensible action would be to encourage a “catch and eat the tasty foreign buggers” policy.