There were a few companies that did conversion kits to basically change your No4 Lee Enfield into the upgraded No9 like the military did. Sterling were quite famous for them. These changes consisted of a new heavy target barrel, a new bolthead to hold higher pressures, new extracter and ejector to deal with the new rimless cartridge, new magazine and an insert for the charger (stripper clip).
Others upgraded such as a Canadian target rifle though it used the original magazine so was single shot, the No9 Lee Enfield which were upgraded various No4 models. There was the lovely L39 and L42 target and sniper rifles (I would love either of these but they are way past my price range) and a number of others,
The No1 MkIII could not be upgraded however as it used a weaker much thinner reciever and other parts. The Indians however did come up with a new version of the basic design using 1950s metalurgly that was strong enough to hold the higher pressures and we got given the 2a and 2a1. There about the only Lee Enfield in 7.62mm nato that you going to find for a decent price and lack the refinement and finish of the others.