There won't be new editions for several years apart from the obvious boxset with all the PJ films, theatrical and extended, and no 4K edition will come until the LOTR trilogy has been re-scanned at 4K and the effects are redone in 4K, which won't really be practical until after the end of the Avatar film series, which WETA will be making their bread and butter from over the next several years and that will be done in 4K.
I doubt they will go back and redo the effects in 4k as that would cost millions of dollars. They'll just upconvert the 2k or 2.8k DI's as they have with other supposed 4k masters... unless a movie's film elements solely rest in the "analog" celluloid world, which would then allow for a fresh, new 4k master without upconverting anything.
I think at some point they will redo the digital effects in 4K and re-transfer the original film elements in 4K (or maybe even 8K), but I don't expect that anytime in the next 5 years at least. Maybe in 2021 for the 20th anniversary or something.