On a Purdey, would a firing pin for a 12 gauge also fit a 20 gauge?

To be honest, I don't know for sure but my feeling would be that because of the scaled action the 12 would be significantly longer than a 20. However that said, I wouldn't be surprised if they are not the same diameter and profile so might be adaptable. After all they are subjected to very similar forces so making the 20 slimmer would make little sense.

As most firing pins need to be adjusted to the particular gun for its length ,throw and end float .Some times a slight difference in one of the diameters can cause problems .
All you can do is to try it in and see .