Scotland Yard has dropped an investigation into the comedian Jo Brand, who joked about throwing battery acid, rather than a milkshake, at rightwing politicians on a BBC radio programme.

Officers said they would take no further action on Friday evening. Earlier, Brand had apologised for the comments she made on Radio 4’s Heresy show.

She received some support, however, from fellow comedian David Baddiel, who attacked the BBC for editing the remarks out of a repeat.

“I don’t think I would have nipped it out. Morally wrong? I’m not sure. I think they’re just trying not to cause trouble,” he told BBC’s Newsnight programme.

Baddiel, who created Heresy, said: “The BBC are still, to some extent, the Auntie of the nation and they don’t like trouble. Even though they did commission a show, Heresy, that was designed to push the boundaries of what people might think and say.

“If it was up to me, I would have kept that line in for the repeat. Apart from anything, it’s a bit silly when it’s had massive coverage to cut it out – that looks a bit cowardly.”

Brand’s joke on Tuesday was in response to a question about the “terrible time” politics was experiencing.

She said: “Well, yes, I would say that but that’s because certain unpleasant characters are being thrown to the fore and they’re very, very easy to hate and I’m kind of thinking: ‘Why bother with a milkshake when you could get some battery acid?’”

Immediately afterwards, Brand clarified her remarks, adding: “That’s just me. I’m not going to do it, it’s purely a fantasy, but I think milkshakes are pathetic, I honestly do, sorry.”

Nigel Farage, who had milkshake thrown at him during campaigning for the European elections, demanded the police take action against Brand, accusing her of “incitement of violence” over the joke.

But remain campaigner Femi Oluwole and others accused Farage of double standards, citing a comment he had made in a speech in 2017 saying he would be forced to “pick up a rifle and head for the front lines” if Brexit was not delivered.

Far-right campaigner Stephen Yaxley-Lennon, also known as Tommy Robinson, also had milkshakes thrown at him during his unsuccessful bid to become an MEP.

On Friday night, Scotland Yard said: “Police received an allegation of incitement to violence on 13 June, relating to comments made on a radio programme.

“The referral has been considered by the Metropolitan police service and no further police action will be taken in relation to this allegation.”

It is understood the complaint to police did not come from Farage or from his Brexit party.

Earlier on Friday, the Sun newspaper quoted Brand as saying the comment was probably “somewhat a crass and ill-judged joke that might upset people”.

She added: “Of course, I’d never do anything like that. It was purely a fantasy. I’m sorry.”

Brand also said she did not mention Farage by name and said female politicians and public figures have regularly faced “far worse” threats.