I was just at the NRA show in Houston and got to talk extensively with a Hornady engineer. One of the questions I asked him--he was pretty much an oracle of knowledge--was about using modern lead-free bullets in pre-WWII rifles. Obviously, what he said applies in spades to pre-1900 guns. My hunting buddy, who shoots strictly modern factory rifles, has a passion for these hi-tech bullets and has tried to get me to use them too. So I asked, and this is what the engineer said.
In a word: DON'T (well, that's actually two words, but hey).
1 - Accuracy(a): lead-core bullets--especially those with an exposed base--expand to fill the grooves quite easily under the relatively low pressure of cartridges like a .303. Not so all-copper bullets. Therefore, lead-core bullets are somewhat more forgiving about groove-diameter variations that you may find in older rifles. I don't know if you had your Martini slugged, but chances are the groove diameter is greater than .311" or even .312." The Martini Enfield was meant to shoot black powder, which fouls and constricts the tolerances within the barrel--and some makers compensated for this by producing a slightly larger bore.
2 - Accuracy(b): copper has a much lower specific weight than lead. Therefore, what copper loses in specific weight it needs to compensate in length in order to reach the weight of an equivalent lead bullet. And bullets longer than X (X being the maximum length at which they can be stabilized given your rifle's twist rate) will tend to tumble out of the barrel.
3 - Pressure: copper being much less ductile than lead, in order for the bullet to be swaged down in the forcing cone, a great spike in pressure will be produced. That pressure may often be excessive for older steel, which has (among other things) a much lower tensile strength than modern. This may not have anything to do with accuracy per se, but it may create bulges or even ruptures in an older barrel.
Therefore, if you have an older rifle it is better in terms of safety, performance, and the firearm's integrity to stick with lead-core bullets. Also, many of these all-copper projectiles are designed to perform best at much higher velocities than those typically found in 19th century military loads--which were the ones taken to Africa and the ones that made people like Bell legendary (he used the old 215gr .303, the old 173/175gr 7x57 as well as the old 160gr 6.5x54MS). That "magical" velocity range between 2,000 and 2,400 fps greatly reduces the need to employ premium, hi-tech bullets and any good welded-core jacketed lead will perform just fine on soft-skinned game.
I hope this helps.