However, Labour will be disappointed in its overall vote, especially in Scotland where again the Conservatives made gains, albeit from a low level of existing support.
Significantly, former Ukip backers appeared to have swung behind her call for a "mandate for Brexit".
"Currently, we don't have them anywhere near that kind of figure - hence a reduced estimate of what their majority would have been if this had been a general election", Professor Thrasher added.
Early results from the polls showed that the ruling party gained more than 550 seats and control 11 councils, while the main opposition Labour Party lost more than 380 seats and control of councils in its traditional strongholds, such as Glasgow.
John Curtice, a professor of politics at Strathclyde University, said the Conservatives were on course for their best local-election result in at least a decade, and possibly in a quarter century.
"We know we ahve worked really hard, we have done an open and honest campaign It is not for a lack of a credible policies". "But we, the whole Labour movement and the British people, can t afford not to seize our moment".
UKIP lost their two seats on the council while the Liberal Democrats held on to their 13 seats. "I will not take anything for granted and neither will the team I lead, because there is too much at stake", she said.
"The reality is that today, despite the evident will of the British people, we have bureaucrats in Europe who are questioning our resolve to get the right deal".
"I want to use this opportunity thank every Labour candidate and every party worker and supporter for the incredible effort they've put in in the past few weeks in this election".
With results declared in 26 councils, UKIP had lost all 30 seats it held and failed to win a single contest.
Arron Banks, the donor who bankrolled Ukip's growth in recent years, said the party was "finished as an electoral force" and its leadership needed "a strategic bullet to the back of the head".
Labour took a battering, despite some high-profile victories in English mayoral contests and smaller than expected losses in Wales. By mid-afternoon on May 5th the Tories had net gains of 481, Labour net losses of 321, the Lib Dems net losses of 44 and UKIP net losses of 131.
In his interview with the Standard, Mr Khan, who is seen by many as a future Labour party leader, also vowed to "stand up" for Londoners against income tax hikes after Labour hinted it could target those earning more than £70,000 and expectations the Tories could drop their "triple tax lock".
Voters turned out on Thursday (May 4) to elect thousands of local-level government seats throughout Scotland and across parts of England and Wales, as well as a handful of newly created regional posts.