Your bolt face and extractor won't work work the the larger and rebated rim of the .308.
A few things:
* 22-250 and 308 will fit in the M77 short action just fine.
* 22-250 and 308 have the same rim size, head size, etc, so bolt face, extractor, etc, are all fine.
* 308 does
not have a rebated rim.
* It will work perfectly well.
I would call the folks at McGowen Barrels or Shilen Barrels. They can thread, profile, chamber and crown the barrel, and screw it in and blue it for you. They can also true your M77 action and bolt for a very reasonable price. Your rifle will shoot much better than if you just take it to a local smith and ask him to screw a new barrel into it.