A Lee-Enfield in .303 British is absolutely the wrong place to start for a conversion to .300 Winchester Magnum. The action is neither long enough, nor strong enough to handle magnum pressures.
The "Enfield" action mentioned is the P-14 in .303 caliber, or it's US version called the M1917 "Enfield". This is truly a magnum length action, but converting it to a sporter is a lot of work and expense. Work needed includes: opening up bolt face, grinding off rear sight ears, drill & tap for scope mounts, bending bolt handle, rebarreling, and restocking, etc.
The late, great P.O. Ackley wrote that he did not recommend rechambering the original .30-06 barrel to .300 Winchester magnum because the chamber area was rather thin, and he had encountered conversions with bulged chambers.