I kept the number plates, chassis plate, and cut the VIN number out of the chassis.
When I registered my car the DVLA took the details of the donor and removed it from their records.
Just remember to keep SORNing the donor until you're done, and make sure the DVLA do actually remove it. I had to remind them..
You can't re-use the chassis number from the donor (that's called ringing!) - you'll need to contact the DVLA to get a new chassis number to use.
Not sure how this works now the local offices are all closed though.