Prime Minister Theresa May's party gained more than 300 extra council seats and at least two mayoralties in England, mostly benefiting from UKIP losses.
May said she was "encouraged" by the results, but said it was important to keep fighting for votes. "I have to say I'm delighted, the effect is to galvanise the SNP support".
"It's encouraging that we've won support across the whole of the United Kingdom but I will not take anything for granted and neither will the team I lead, because there is too much at stake", she said.
Shadow chancellor John McDonnell said the results were "not the wipeout that some people expected", which is not the most convincing defence ahead of a general election.
"... as the local elections showed, they are ... are on course for the type of victory last seen by (Labour's) Tony Blair and Margaret Thatcher when they were at their peaks", said Adrian Drummond of pollsters Opinium.
The Tories gained over the Opposition Labour party in numerous council seats and confirmed pre-poll forecasts of a Labour battering, largely blamed on the leadership of Jeremy Corbyn.
After a record result in the Scottish local government elections, where they returned 276 councillors, the Conservatives now have seats such as Mr Salmond's Gordon constituency and the Moray seat of SNP depute leader Angus Robertson in their sights. "Only a general election vote for the Conservatives will strengthen my hand to get the best deal on Brexit".
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Top EU officials say they want liberal independent Emmanuel Macron to beat his nationalist rival, Marine Le Pen. At a press conference in Brussels, Mr Tusk said: "These negotiations are hard enough as they are".
Meanwhile, Labour slumped to become the third largest party in Scotland's councils, and was kicked out of power in its Glasgow heartland for the first time in nearly 40 years.
"Tory MPs from Scotland will be a rubber-stamp for Theresa May, for her reckless approach to Brexit and the cuts we know the Tories will want to impose".
Paul Nuttall's Ukip was crushed as every single councilor facing election suffered defeat.
The former Ukip MP Douglas Cars well, who defected from the Conservatives, said it was "over" for a party that once soared to overall victory in European elections in 2014 and secured 4m votes in a general election under the stewardship of Nigel Farage. "But we, the whole Labour movement and the British people, can't afford not to seize our moment".
"The local elections proved that the Labour Party is finished and the Conservative Party is heading for a landslide", he said.
"The Liberal Democrats are the only party that can challenge Theresa May".
The Lib Dems increased their share of the vote by 7% but lost more than 40 seats. As long as Jeremy Corbyn is in power, they say, Labour will not be receiving their vote, as the Chippenham native just does not "speak for them". "We will be the strong opposition that will stand up for you".
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