Like I said, studies shown that having an ID didn't change much. Plus, they already can! Commonwealth citizens can vote in UK elections pretty much as soon as they arrive.
I question the results of these studies. I think it's very difficult to measure empirically how many people are cheating if it is not required to have voter ID / some sort of proof of your identity when voting.
Presumably at least a few of them anticipate the easy problems and design a methodology appropriate to dealing with them.
For instance, if voter ID is highly effective, you'd expect much higher rates of double votes under a given registration in places that don't require it (unless the cheaters are masters of anticipating registrants that aren't going to vote).