As Labour fell to third place, deputy leader Tom Watson said: "We need a change of direction urgently".
Meanwhile, shadow foreign secretary Emily Thornberry said she wanted the party to campaign to overturn the result of the 2016 poll.
Over on Labour's side its evolving new position on whether to hold a second referendum on whatever Brexit deal is agreed is coming under fire.
With some results still to declare, Labour was on course for less than 15% of the vote - worse than the party's previous low in 2009 - and a fifth place finish in Scotland.
He added: "Normally elections are won by the party with the highest number of votes and the most seats, at least that is what I thought until this morning when they tried to reinvent all of this".
He will add: "If we've learnt anything over the past year it must be that we now know that we can not enjoy all the rights of being a member of the European Union with none on the responsibilities that necessarily go with it".
Esther McVeyLeadership candidate Esther McVey said that Brexit would happen at the end of October either with a new Brexit deal, or without a deal altogether.
When asked about Boris Johnson, who is now the odds-on favourite to take over from Prime Minister Theresa May in the current Conservative leadership contest, Mr Farage said that though he personally liked Mr Johnson, his record on Brexit was questionable, citing the MP backing Mrs May's widely unpopular withdrawal agreement.
Former front bencher, Lisa Nandy, says promoting a second referendum will be seen as a "final breach of trust" in her Leave-backing constituency of Wigan.
People's Vote campaigner Alstairs Campbell, who voted for the Liberal Democrats for the first time, repeatedly tweeted that Mr Farage had lost.
That said, Johnson could risk triggering another vote of no confidence and losing his job as PM if he insists on pressing ahead with a no-deal Brexit in order to please the Brexiteers.
Lord Heseltine admitted he voted Liberal Democrat at the recent European Parliament elections and had the Tory whip removed beforehand after he stated his intention to do so.
After May announced on Friday that she was stepping down, many of her would-be successors have said they wanted Brexit with or without a deal, rejecting another public vote.
As the leadership race got underway, the contenders began setting out their stall with policy announcements.
It is not clear where candidate Sajid Javid would put himself on the Brexit spectrum, but the former banker had originally backed the remain position before becoming pro-Brexit.
Labour chairman Ian Lavery told BBC Radio 4 that the results were disappointing, but that the Conservatives performed worse, suffering a "whiteout".
"I want a leader who is going to be straight with the British people, straight with the Conservative Party about the choices and options that are available".
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