In his judgment, Mr Justice Foskett rejected a submission by the MoD, which denies negligence, that all the cases were "doomed to fail" on the issue of causation.
He refused to strike the cases out and said the nature of the injury or disability in question was an issue of fact that only the judge who heard the full trial could determine after having heard all the evidence.
Dead or untraceable
He said: "All things being equal, a veteran who believes that he has an illness, injury or disability attributable to his presence at the tests whose case is supported by apparently reputable scientific and medical evidence, should be entitled to his 'day in court"'.
The judge acknowledged that it would not now be possible for the MoD to call as witnesses many of those responsible for the planning and execution of the tests.
The MoD's counsel, Charles Gibson QC, said that over 90% of the 114 essential witnesses were dead or untraceable.
Speaking after the judgment, veteran Alan Ilett, 73, from Chelmsford, said: "It's a good judgement. I'm giving a thought for those poor souls who didn't make it to here."
Mr Sampson, senior partner at Rosenblatt Solicitors which acted for the servicemen, said it was a "wonderful day for everybody", and added that "since the MoD started the action 59 of the clients have died".
He said: "We still have a further period of perhaps three years before the case can finally be brought to court for trial and sadly, in that time, many of the veterans we are fighting for will have passed away.
"We hope that the Ministry of Defence will recognise this and agree to settle the claims of the veterans out of court, rewarding them with the compensation they rightly deserve."
During the hearing in January, Benjamin Browne QC, representing the ex-servicemen, said science had made a link between health and their role in the tests.
He said the UK government's attitude contrasted with those of many countries around the world who had set up schemes to compensate veterans as they fell ill.
Nuclear testing was carried out on Christmas Island in the South Pacific
The US has awarded compensation under the Radiation Exposure Compensation Act to veterans, including at least one Briton, involved in nuclear testing in the 1960s.
In France, a government-backed bill that is expected to pass later this year would provide compensation to those who contracted illnesses attributed to the country's nuclear tests in the Sahara and French Polynesia between 1960 and 1996.
At the height of the Cold War in the 1950s, Britain carried out a series of nuclear weapons tests in mainland Australia, the Montebello islands off the west Australian coast and on Christmas Island in the South Pacific.
Veterans who served in the Army, Royal Navy and Royal Air Force, as well as personnel from New Zealand and Fiji, were involved in the tests.
In January, veteran Dougie Hern, 72, told the BBC what happened.
"We saw a bright, brilliant light," he recalled. "It was as if someone had switched a firebar on in your head. It grew brighter and you could see the bones in your hands, like pink X-rays, in front of your closed eyes."
Mr Hern, now 72, believes radiation exposure on that day and four others accounts for his diabetes, the spurs growing on his sternum and the death of his 13-year-old daughter from cancer.
When the hearing opened, an MoD spokesman said it recognised the "vital contribution" the men played.
"When compensation claims are received they are considered on the basis of whether or not the Ministry of Defence has a legal liability to pay compensation. Where there is a proven legal liability, compensation is paid," a spokesman said.
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