Following a hung parliament in the snap election she called, Mrs May has said she plans to form a minority government to deliver Brexit and will take part in the negotiations on 19 June.
“This Government will guide the country through the crucial Brexit talks that begin in just 10 days and deliver on the will of the British people by taking the United Kingdom out of the European Union,” she said outside Downing Street.
As he congratulated Mrs May, Mr Tusk said in a letter: “Our shared responsibility and urgent task now is to conduct the negotiations on the UK’s withdrawal from the European Union in the best possible spirit, securing the least disruptive outcome for our citizens, businesses and countries after March 2019.
“The timeframe set by Article 50 of the Treaty leaves us with no time to lose.”
Earlier, Mr Tusk warned Britain might end up with no deal on its withdrawal arrangements or future trade if it did 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’.”
Brussels’ chief negotiator Michel Barnier earlier said Brexit talks could wait until the new government was ready.
He said on Twitter: “Brexit negotiations should start when UK is ready; timetable and EU positions are clear. Let’s put our minds together on striking a deal.”
His comments suggest the EU is willing to delay the start of official negotiations, which were due to start on 19 June.
Guy Verhofstadt, the European Parliament’s Brexit co-ordinator, described the General Election result as “yet another own goal” for the UK.
The former Belgian prime minister wrote on Twitter: “Yet another own goal, after Cameron now May, will make already complex negotiations even more complicated.”
Scotland’s First Minister Nicola Sturgeon has called on all parties to “join together” to keep the UK and Scotland in the single market.
Meanwhile, the chair of the European Parliament’s centre-right grouping, Manfred Weber, warned that “the clock is ticking for Brexit”.
“Therefore the UK needs a government soon. The date for the beginning of negotiations is now unclear,” he wrote on Twitter.
“Our position is clear: we want good relations with the UK but Brexit means leaving the EU and the advantages that go with it.”
Ireland’s prime minister-elect Leo Varadkar said the election result was an opportunity for Ireland.
“We must ensure that the Brexit talks are handled in a smooth and coherent manner to secure the best possible outcome for Ireland, for Europe and the UK,” he said.
“The results of the UK election indicate to me that there is no strong mandate to proceed with a hard Brexit, which represents an opportunity for Ireland.”