The interviewer was also a respected current affairs reporter who fronted the award-winning 1970 documentary Gale is Dead, about the life of a young drug addict who died of an overdose at 19.
Rantzen, who fronted That's Life, paid tribute, saying Williamson was "one of the most talented interviewers of his day".
Williamson, who lived near Frimley, Surrey, died on Sunday after a short illness.
"He became immensely popular with his interviews for Children Talking on Braden's Week," Rantzen added, referring to a show he worked on before That's Life.
'Magic'
"No one else could interview children like he could. He will be greatly missed and long remembered."
Williamson's career began when he joined the BBC as a freelance reporter in the north-east of England in 1957.
He went to appear on Voice of the People in which he talked to ordinary people about issues in the news.
He moved on to working with children, encouraging them to chat on camera.
Describing his work with children, he said: "The magic is that kids are so unpredictable. Everything is possible in their world."
He was also a reporter on the BBC Two current affairs flagship Man Alive, which was edited by Rantzen's husband Desmond Wilcox, who died last year.