The error occurred on Tuesday this week during an exclusive appearance by the Wombles on Simon Mayo's Drivetime show to perform various songs, including new single 'Wombling Merry Christmas'.
Some parents were outraged when Orinoco took off the head of his costume while the web stream was still running and available on the Radio 2 website.
Wombles inventor Mike Batt, who was playing Orinoco, removed the head for the interview with Mayo at the end of the appearance, shattering the illusion for children that the Wombles were real.
The BBC quickly removed the link to the live feed from the Radio 2 website and Batt apologised on Twitter for the incident.
According to The Daily Telegraph, parents have complained of the 'damage' done by the gaffe, forcing them to come up with explanations about why a human was inside a Womble.
Peter McFarland, a dad of four who was watching the live stream with his 6-year-old son Dylan, told the paper: "I'm very very angry - we were watching the Radio Two live footage of the Wombles and at the end of the interview Orinoco pulled his head off before the cameras stopped rolling to reveal Mike Batt instead.
"Some idiot at the BBC forgot to switch off the webcam at the end of the show and Dylan was right in front of the computer."
He added: "I thought it might have been too brief for Dylan to have noticed as he went totally silent and we thought we might have got away with it, but then he said 'all Wombles are fake', and asked to go to Wimbledon Common to meet the real ones.
"Now I have an absolutely crushed and distraught 6-year-old. One VERY angry email getting sent to the BBC now."
Christine Furniss, 43, said that she had to explain to her two boys that the Wombles were "not real" after the live stream error.
"They both looked at me for an explanation when Orinoco's head came off - what could I tell them? I had to tell them the truth and it's fair to say they were devastated," she said.
"All of their friends were watching it at home too as it was a chance to see the Wombles perform the Christmas song - so a lot of children were left very, very upset about what happened."
Batt, who created the Wombles in the 1970s, claimed that that BBC had told him he was 'no longer on air' and could remove his costume.
He also said that he was supposed to be given a break between the performance and being interviewed by Mayo as it was very hot inside the studio.
He said: "I told [the BBC] that I needed a break after the songs [as it] was a high heat effort - they were supposed to give me a two-minute break!"
But a BBC spokesman claimed that Batt was made aware that the Wombles were being filmed, and said he "had not been given the all clear that they were off air".
"The Wombles played a festive set for Radio 2 which was filmed for live online streaming. The Wombles were aware they were being filmed, but [Mike Batt] just took his head off," he said.
"The group were fully aware that they were being filmed and had not been given the all clear that they were off air."
Last month, the UK advertising watchdog ruled out an investigation into a Littlewoods Christmas TV ad campaign, despite receiving more than 450 complaints that it distressed children by suggesting Father Christmas is not real.