CT KID;
As I recall, the 7mm-08 is made by necking the .308 Winchester case to hold a 7mm bullet, with no other changes. The 7x57 case has more taper than the .308/7mm-08. In a bolt action rifle, the chamber would have to be cut and the barrel set-back one full thread to hold the new chamber.
If that is true, then your rechamber idea would NOT work on the NEF Handi rifle, because the barrel can not be set back.
HTH
John