Which Enfield?
SMLE: Not much scope for change to a current caliber. There were all sorts of British sporting rounds used in them, but nothing common today. The Ishapore is supposedly made of better steel to hold up to .308.
No 4: The British converted a lot of them to .308 as a stopgap measure. Don't know if you can get the parts to DIY, though.
Gibbs is making them into .45-70s and I have heard it is easier to set one up in .405 Winchester... which is getting to be available again.
Pattern '14: You can turn one of these into about anything, I figured that is what you had when you mentioned .338. But it is a lot of work; probably cheaper to buy a new gun than to pay a gunsmith to rebuild a milsurp that much. Great DIY if you want to get into it.