Can he simply ream a 6mm Remington chamber?
No. Sorry. Except for the case length and case neck diameter, the 6mm Remington is larger than the 6mm Lee Navy,
If the barrel has enough length in the shank, you could cut the 6mm Rem chamber off, re-thread the barrel, then ream a new 6mm Lee chamber, But you'd probably be better off starting with a 6mm barrel blank.
You would also most likely have to do some work to the action, bolt face, extractor, magazine rails, etcs...
A rifle chambered in .220 Swift would be your best bet, .225 Winchester might work reamed out, might have to set the barrel back a couple threads....
However, even with those, you're going to have to deal with the 6mm Lee round's rim being approx .03" smaller in dameter than the Swift's rim.
If you don't have 6mm Lee brass (and who does????
) you could pobably form a workable case from .220 Swift brass (good luck getting those, as well..) cases will be a bit short, but should be shootable in a bolt gun.
Sorry I don't have any better advice.
Good Luck with your project!