Speaking at the UKIP conference in Torquay, Mr Bolton urged party members to "rally around the party", saying: "Without being united, we can not lead". Instead, somewhat surprisingly, the party members went for a former Liberal Democrat.
Henry Bolton has already stressed he will seek to unite Ukip after defeating a candidate whose description of Islam as "evil" had threatened to split the party.
Former UKIP leader Nigel Farage, who led the party's successful Brexit campaign, had warned UKIP would be "finished" if it embraced an anti-Islam platform.
He has said he will have conversations with Ms Waters, Mr Farage and Mr Banks about their futures with the party.
A former soldier and police officer, Bolton won a tight seven-way race, taking 29.9% (3,874) of the overall votes.
"The other thing is, I do not believe for a moment that we should be pushing this out and allowing the negotiations to be endless before we leave".
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Speaking at the UKIP conference in Torquay, he said: "Brexit is our core task however, it is not the end of the line".
Insiders say if the Pegida UK co-founder and anti-sharia-law campaigner Anne Marie Waters is selected, the extreme views of her followers could push the party over the cliff edge.
"If you look at the things I've done, you'll see a certain degree of endurance there, both physical, emotional and political", he said.
Bolton has become the fourth leader of UKIP in the last 12 months.
Henry Bolton has been elected the new leader of UKIP in a surprise result.
Asked at a press conference after the result was announced whether this made him an "establishment mole", Bolton said his background made him ready for the job.
"I had expected to win", he said, while carrying what appeared to be a copy of an unused victory speech containing the headline "New World Order". She resigned after just 18 days in the post, saying she could not work with the party hierarchy.