Fortunately the barrel shank and thread pitch are the same for the M77 and No 1 rifles. But the shank length on the No 1 is longer, and there is an ejector cutout for the No 1. So a gunsmith could extend the shank length, clock the barrel to the receiver to align the sights, then cut the ejector cutout. Now you need to think about the forearm...if I remember correctly the 338's were 24", which may be a different profile than the current No 1 barrel.
Hope this helps,
Ted