I believe this is supposed to take place five years after the second movie, since it's mentioned that the old Uni-Sol program was closed down "five years ago". Since then, a new Universal Soldier that has been created, one that is augmented with gene therapy, is immune to high radiation and whose body parts can be replaced by cloned parts.
In the opening sequence, which takes place in Eastern Europe, a brother and sister are kidnapped by one of these soldiers and taken to a hideout in the abandoned city of Chernobyl, where the leader wants the freedom of political prisoners or else he'll explode a bomb that will release a radioactive cloud into the air.
When Jean Claude Van Damme shows up as Luc Devereaux he's being cared for by a doctor who is trying to re-train him to be a human. The entire story of UNIVERSAL SOLDIER: THE RETURN never happened. He didn't marry that reporter and didn't have a daughter. Here, he's this damaged person. And it's mentioned it's only two years after the events of the first movie (1992)--and he's rapidly aging, since he's getting arthritis and looks twenty years older. They need him to go against his rival from that first movie so they re-make him into a Unviersal Soldier and the two fight for the last half hour in this abandoned factory. Then, when he wins that battle, he has to go fight that new Universal soldier.
Of course the ending leaves it wide open for a sequel, which will probably involve clones. But the last thing I want to see is another UNIVERSAL SOLDIER movie....this one sucks big time.