Results showed a loss of 144 seats for Labour - only five weeks before the snap general election.
Despite the thumping victory, Mrs May said there was no room for complacency ahead of the June 8 general election and the negotiations that follow on Britain's withdrawal from the European Union.
Across Scotland, the Conservatives had the biggest gains, up 164 seats to 276 - pointing to a revival for the party which has only one MP north of the border.
"We cant assume that what happens in local elections will automatically be replicated in general elections", said Tory minister Brandon Lewis.
"We lost seats but we are closing the gap on the Conservatives", he said.
May has a working majority of less than 20 seats in the 650-seat parliament but polling analysts have predicted she could increase that number by as many as 100 at the June 8 election.
'But we, the whole Labour movement and the British people, can't afford not to seize our moment'.
The British people have been held back for too long. In Derbyshire, the Tories took 19 seats from Labour as the county passed directly from one party's hands to the other.More news: Portugal Wins 2017 Eurovision Song Contest
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Conservatives won the West of England metro mayor contest, while Labour claimed the mayoralties in Greater Manchester, won in the first round of voting by former cabinet minister Andy Burnham, Liverpool, Doncaster and North Tyneside. What we have to do is make this election about more than leadership, we've got to make it about the future of our country.
Speaking during an early-morning round of interviews, he said: "So far, it's been a tough night, but it hasn't been the wipeout that people expected".
Cable said: "Ukip has virtually disintegrated, and that matters politically because, if you are a middle-of-the-road Conservative who voted Remain and you see your own party effectively assimilating Ukip and its values as well as its voters, I think they will have pause now and worry about the future". The party gained control of 11 councils for a total of 28 and added 559 new councilors across England, Scotland and Wales.
"We've really worked hard as councillors in the Labour Party to ensure that our candidates for the council have come forward as well".
The party's success in Glasgow forced Labour out of power in the city for the first time in nearly 40 years.
The Lib Dems had 441 councillors, a net loss of 37; the Scottish National Party had 431, up 31; Plaid Cymru was on 202, a net gain of 26 and the Greens were on 40, a net gain of six.
I urge everyone to vote Labour because things can, and will, change.