Update 11.20am: Talks on Britain's withdrawal from the EU will begin when the new UK government is ready, Brussels' chief negotiator Michel Barnier has said.
The comment suggests Mr Barnier is ready to delay the opening of official negotiations, which were expected to begin on June 19 in Brussels.
Meanwhile, European Council President Donald Tusk warned of the risk that Britain might end up with no deal on its withdrawal arrangements or future trade because it does not open negotiations in time.
"We don't know when Brexit talks start. We know when they must end," said Mr Tusk. "Do your best to avoid a 'no deal' as result of 'no negotiations'."
The president of the Socialists and Democrats grouping in the European Parliament, Italian MEP Gianni Pitella, said: "It's a disaster for May. Her huge gamble has backfired spectacularly. She has no credibility in UK or Europe. She should resign.
"May's wasted enough time. Negotiations must start. We want a positive relationship but rights and benefits can't be the same for UK post-Brexit."
German foreign minister Sigmar Gabriel said he hoped the election result would be seen as a message that the British people do not want a hard Brexit.
He urged the new government to conduct "fair talks with the European Union and reconsider whether it's really good for Great Britain to withdraw from the European Union in this way".
And he said he hoped the new government was one "with which we can conduct serious negotiations and if possible keep Great Britain as close as possible to the European Union".
Update 8.45am: Nigel Farage believes Paul Nuttall should face no blame for Ukip's woeful election showing, as he left open the possibility of a return as leader.
The MEP said he would have "no choice but to return to frontline politics" if the Brexit he desires is put at risk by a hung parliament.
But the tensions within Ukip following the collapse of its vote were clear to see as MEP Bill Etheridge claimed Mr Nuttall had surrounded himself with "appallingly bad advisers".
He added "heads must roll to save the party" and it "feels like a dictatorship and the leader is not the dictator".
Mr Nuttall failed in his bid to win a Commons seat, finishing a distant third in Brexit heartland Boston and Skegness behind the Tories.
Elsewhere, the Ukip vote seemed to disappear, splitting between Labour and the Conservatives.
Former party leader Mr Farage told LBC Radio: "I don't think Paul did anything wrong at all, he didn't have time to establish himself with the public."
He added: "I would not lay any blame on Paul Nuttall whatsoever."
Asked if he would return as leader, Mr Farage said: "Whether leading or playing a prominent role is perhaps a different question."
He added that he could not see the "historic victory" of the British people at last year's EU referendum be betrayed by the political class "and not fight back".
Mr Farage also joked that European Commission president Jean-Claude Juncker will be on his "fifth bottle" of wine following the result.
He said "They will be cheering in Brussels" as they see a coalition under a weak leader, adding: "They will think this is the chance to crush a real Brexit."
Mr Etheridge, who also came home a distant third in Dudley North, labelled the election a "disaster" for Ukip.
He said: "These advisers have led the party down a route of authoritarianism and an obsession with Islam, which has turned off many voters, when we should have been focusing on Brexit."
Mr Etheridge went on: "Paul was overwhelmingly elected as Ukip leader because the membership believed that he could unite the party and lead us on to more success.
"But what we have seen is three electoral disasters in a row with the same people at the top, refusing to learn from their mistakes."
Mr Nuttall's humiliation comes after he failed to win in the Stoke Central by-election in February.
An indication that it was going to be a difficult night for his party came in Sunderland, which a year ago provided an early sign that the European Union referendum was going to result in Brexit.
In Houghton and Sunderland South, Ukip's vote was down 15.78%, in Sunderland Central it fell 14.25%, and in Washington and Sunderland West it was cut by 12.85%.
Earlier: Europe's Budget Commissioner has warned that a weakened Government in London could undermine the Brexit process.
Gunther Oettinger told German radio station Deutschlandfunk that in negotiations, "a weaker partner weakens the whole thing".
If both sides in a negotiation were strong, "you get results more quickly", he said.
Mr Oettinger said that the European Union was ready for Brexit talks which are due to begin on June 19, and would be "hard but fair" in its negotiations with the UK.
But he warned that time was running out to reach an agreement which could be approved by all 27 remaining governments within the two-year deadline set out in Article 50 of the EU treaties.
Theresa May's decision to hold a snap election had delayed matters, he said, adding: "There is only 15 or 16 months left."
Mr Oettinger said that he did not believe that voters made their decision on the basis of Brexit, as Mrs May hoped.
"It was about social justice and security," he said.
Czech prime minister Bohuslav Sobotka said he was "delighted" that Jeremy Corbyn had significantly strengthened his party's position, saying that the Labour leader was "the real winner of the British election".
Mr Sobotka said that Europe would now have to wait for the formation of a UK Government, but said he did not expect it to take long.
"Britain launched Brexit in March," he said in a tweet. "We only have two years."
An article on the Europe edition of website Politico, which covers the politics of the European Union, said that Britain was waking up on Friday morning "more divided and uncertain about its future than anyone thought possible".
The article by Tom McTague added: "From a position of relative strength, dominating a compliant parliament which had accepted Brexit, Theresa May is now struggling to cling on to her job, unsure whether she will even be able to form a minority government."
The article said Mrs May is "mortally wounded" and looked ahead to who could potentially emerge from the political wreckage, saying: "Boris Johnson is the most obvious Tory winner from the fallout.
"When a steady but uninspiring leader has been found wanting, they may turn to a tried and tested winner with the charisma to take on Corbyn."
Manfred Weber, chairman of the European People's Party (EPP), the largest grouping in the European Parliament, said Britain looked disorientated by the election result.
The German MEP told BBC Radio 4's Today programme: "One year after the decision by the British people to leave the European Union, we see that Europe, Paris for example, Berlin and even Brussels is very stable, so we are ready, and we see disorientation in London, which is not a positive thing."
Asked whether he expected to negotiate with Theresa May, he said it was purely a domestic matter, adding: "We want to start, the time is running, and instability, losing time, is not in your or our interest.
"Europe is for the moment strong and united and we are waiting for Britain."