To all of you guys saying that removing a barrel is somehow required for a quality job....................hogwash!
I have and do many rifle re-chamberings without removing the barrel from the action, belted magnums being the easiest, generally.
The OP, the Weatherby chambering will completely clean up a Winchester chamber, the headspace will stay exactly the same if whoever does the job measures correctly, it should be .220" from the base to the forward section of the belt.

Cheers.