One of the alleged victims was a resident at the notorious Haut de la Garenne children’s home which was at the centre of a high-profile police investigation into historical child abuse on the island in 2008.
He claimed to have been taken to the island’s main theatre, the Opera House, as a “treat” before being taken backstage to meet Brambell, who he accuses of molesting him in a back room.
The second victim, who had not been a Haut de la Garenne resident, also claimed to have been abused by Brambell at the theatre. The alleged victims were aged 12-13 at the time.
Brambell, who died in 1985, was homosexual and had a criminal record for “persistently importuning for an immoral purpose” in a public lavatory dating from 1962.
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The allegations were made to former Jersey health minister Stuart Syvret, the most outspoken critic of the island’s failure to deal with child abuse when the Haut de la Garenne scandal erupted four years ago.
He was contacted by dozens of people who had been abused on the island, some of whom named Jimmy Savile as one of their abusers.
Mr Syvret told The Daily Telegraph: “Two of the people who contacted me said that they were abused by Wilfrid Brambell.
“I was told that he visited the back rooms of the Jersey Opera House and abuse would happen there.
“I am convinced they were telling the truth. One of the men was a resident at Haut de la Garenne at the time and was taken to the theatre where he was abused.”
Mr Syvret said the two victims had not reported the abuse to the police because they did not trust the island’s constabulary.
Brambell became a household name in the 1960s and 70s when he appeared alongside Harry H Corbett in Steptoe and Son, in which he played a scruffy rag and bone man constantly derided by his on-screen son as a “dirty old man”.
He also appeared in the Beatles A Hard Day’s Night, playing Sir Paul McCartney’s grandfather, after being hand-picked for the role by the singer.
His conviction for importuning, or soliciting, came after he was arrested in a public lavatory in Shepherd’s Bush, west London, in Nov 1962. He was given a conditional discharge and ordered to pay 25 guineas costs.
Brambell had been briefly married in the 1950s but had few close friends or relatives, and when he was cremated in 1985 after dying of cancer at the age of 72, only a handful of people turned out for his funeral.
A spokesman for Ray Galton and Alan Simpson, the writers of Steptoe and Son, said: “There were only about two people at his funeral. He didn’t have much of an entourage and he didn’t really have people who mixed with him socially.”
A spokesman for the BBC said that if the corporation received any allegations about anyone abusing children on BBC premises, it would be investigated as part of the independent inquiry into sex abuse by Jimmy Savile.