This is "The New Wine", a short story by John Christopher (pseudonym of Sam Youd), which was also the answer to this old question. You may have read it in one of these compilations. The SF 68 radio dramatization is available from SciFiMike or the Internet Archive.
It begins with a couple spending their last day together, as the man is going to leave Earth on a spaceship for some research that will take several years. However, while he is going to be only eight years older when he returns, about a hundred years will have passed on Earth.
"There you are," he said. "I suppose we're both stubborn." He laughed. "I'll give your regards to your great-great-granddaughter. Maybe I'll marry her."
She shook her head. In the thickening dusk he could not see the expression on her face.
"That time factor," she said, "is it certain? I don't understand mathematics; to me it seems fantastic."
"I could go over the theory with you, but it would be wasting the little time we have. It's certain enough. The ratio, as far as this trip is concerned, is approximately twelve to one. For us, eight years—for the world we leave behind, a century. We return in late April in the year 2129. That's the nearest our predictions will take us."
"And everyone you know to-day will be dead."
The woman is staying on Earth to help complete a project that will give all human beings telepathic abilities.
"Because it won't be your kind of world. You may return with the prestige of great inter-stellar explorers; they may give you medals and honours; but they won't share their lives with you. They will not be able to, even if they should wish it. To telepaths, a non-telepath will be like something out of the zoo."
He understood now. "Your Project X? It's running smoothly, then?"
"We're building the generators."
Both expect this to be the end of all disharmonies and war, since there will be no more misunderstandings.
The woman thinks telepathy will be a good thing, for reasons not clearly stated; the man has his doubts:
"But to do a thing like that, after only three experiments! And without any reference to the wishes of the people concerned. Aren't you afraid of it going wrong?"
"Have you considered the alternative to a simultaneous planet-wide irradiation like the one we are doing? The principle has been discovered; you can't turn science backwards. The choice is between doing what we plan to do or having the advance take place piecemeal. If we did that, there would be trouble. Resentment of those families with ordinary children against those with telepaths. National resentments, leading perhaps to wars. All the confusion of an interregnum between the old and the new. We shall avoid all that. The world will go forward in one giant's stride."
"After three successful experiments," he repeated.
"You've forgotten the monkeys. There were five monkeys altogether. And the repeatability, from monkey to man, is more than a double confirmation. The point is: it's not something you can do in control experiments over a period of generations. It's potentially dynamite. We believe that the bold course is the right course. The world you come back to will have thanked us for it."
Eight years later - a hundred years later for all who stayed on Earth - the space mission is completed and the crew of the spaceship lands on Earth, only to find that there are no more people. Then the very last survivor finds them. He was the last child born without a telepathic brain.
"The people?" the old man said. "You mean the rest of the people? They died. I had a couple of buddies with me until the hard winter, year before last. That took them. You get lonely by yourself. I wasn't even sure I could remember how to talk. I never saw any sense in talking to myself."
"Harl said: "My God! He was born in the year the Astro went out—before the generators started. Does that mean . . . there was something wrong? It killed them instead of making them telepathic? But the animals? And Whittaker's children . . ."
The old man nodded. "You knew about the generators?"
He is excited, because with the non-telepathic people from the spaceship, mankind could be started anew. But they have only one female crew member and she is 58...
The old man said: "That's why I wanted to live to see you come back. So that things could start again."
They looked at him.
"They called me Lee after the captain," he said. I knew all about the flight. I saw the records. That you had two women in the crew. Things can start again now."
Rennis and Awkright turned and began to walk away.
"Yes," Harl said. "Two. Sub-Navigator Mary Rogers. Assistant Medic Lucy Parino. Aged, respectively, fifty-two and fifty-four."