Yes you can, What can not be done is use the 6mm Remington reamer to clean up the 243 Winchester chamber, the 243 chamber is shorter, BUT! larger in diameter, same for the 308 Winchester when poor planning is used to chamber the 308 W to 30/06. The attempt will leave a portion of the shoulder/case body juncture, When cases are fired in the 30/06 chamber a ring will appear around the case body.
Then there is the neck, if both reamers have the same diameter neck, no problem, if the Ackely reamer has a neck that is smaller in diameter part of the neck in the chamber will have two different neck diameters.
Check the shoulder/case body juncture, the diameter of the case body at the shoulder juncture for the Ackley 30/06 is .455”. the diameter of the case body at the shoulder juncture is .454”. Then there is the neck on the Ackley chambers, it is longer.