What neck size did he ream your chamber to? .313" or so? Most custom chambers have a fairly tight neck.
What kind of brass did you get and how thick are the necks? You may have to turn a lot of brass off the neck in order to have clearance around the neck in the chamber with a loaded round.
For instance in my 280AI with a Hart barrel the neck is .313" and I have to turn the neck brass down to .0125" to get the .003" clearance in the chamber:
with a little bit of springback and irregularities that winds up to be closer to .310". The Nosler brass is mostly .013" to .014" so it is a complete turn
In a 6.5 rem mag Douglas barrel the neck is .294" and the Remington brass has thick necks from .015" to .017" so I have to turn off brass down to .0135" which again is a complete turn.
Same thing in a Brux 338RUM, complete neck turn.
So it will depend upon your neck size and the thickness of the brass as to how much you will turn off.
Checking runout on new cases will not tell you much. Mostly you are fire forming on the first load anyway. You will not be Full Length Resizing, only sizing the necks since the shoulder is way back on new cases and the case body is not large enough to contact the inside of the die anyway.
I would not turn the necks until you know the size of neck anyway. If your gunsmith was on the ball and hadn't already had too much Wild Turkey he should have marked it for you
but if not at least told you.