Cody...
I'm thinking by the time the Browning barrel is modified to fit the Enfield, might be just as good to either start with a fresh barrel...or, and someone can check me on this, see if the existing barrel can be re-chambered for the 300 WM.
Or, if you're only going to burn up a few rounds at the range each year, and a couple more on critters, keep it as it is. Nothing wrong with the .308 Norma Mag, save for ammo is a tad pricey and limited on choices.
Indeed, I know a few here will be horrified that you would want to change it over to 300 WM.
As for restocking it...
Shouldn't be a big challenge to find a good synthetic for that action. However, I'd let a gunsmith advise from his experience on these, and then fit the replacement you buy to make sure everything comes together properly so that you get best performance.
Very good actions, well worth it.
EDIT: Bell & Carlson make a replacement stock for these...
http://www.bellandcarlson.com/
Look under "Carbelite". Note that it specifies "Bent floorplate only", which is the standard bottom metal. Some gunsmiths modify these when customizing, make sure yours hasn't been altered. Again, a gunsmith will know what to look for.
Also, there's some good aftermarket triggers for these that are a HUGE improvement over the original two stage. Well worth it to make this change, too, if it hasn't been done already. Good gunsmith will know what to do in this regard.