During six years in the 1950s on Christmas Island and in the Pacific Ocean, Britain tested nuclear weapons.
Claims by soldiers who took part in those tests that radiation had caused them a series of health problems have won support from the Supreme Court.
It has overturned an earlier appeal ruling. Over 1,000 veterans can now pursue a claim for damages against the Ministry of Defence.
The victims’ lawyer, Neil Sampson, said: “This is a significant step forward in the veterans’ fight for justice but by no means it is the end of the road.
“We welcome the Supreme Court’s decision and hope that we can have a hearing at a very early date but unfortunately it may well be another year before the Supreme Court is able to have a full hearing of this matter.”
One of the veterans, David Harman, said: “I think you have to say would any of those people who are making those decisions go to an island like that or send their children to that island? They wouldn’t, would they?”
The Ministry of Defence said it acknowledged a “debt of gratitude” to those who took part but denies negligence.
The appeal court had ruled the cases had been brought too late to be heard.
As the case has dragged on many of the veterans have since died.