"They're individual contests being fought in very differing situations, from local council-level, issue-driven campaigns up to large mayoral fights with some well-known politicians", a Labour party spokesperson said.
UKIP lost all but one of the more than 100 council seats it held, as voters switched to the Conservatives now that UKIP's main goal had been achieved.
Labour finance spokesman John McDonnell acknowledged the party had suffered a tough night, but told ITV the results were not "the wipeout that people expected".
May has accused European Union officials of seeking to affect the outcome of the election by issuing threats over Brexit, and warned voters that the other 27 member states were lining up against Britain to win a deal that "works for them".
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Despite the apparently dismal showing, Labour MPs have been advised to say the results are merely "disappointing" and that the party has done "better than people were predicting".
Liberal Democrats described early results as "patchy", with the party losing 41 seats and failing to make progress in their targets of Cardiff, Somerset and Gloucestershire.
The General Election on 8 June shows early signs that May will inflict a heavy defeat on her Labour counterpart, Jeremy Corbyn. "We know there is an election taking place in the United Kingdom", Ms Margaritis Schinas said on Wednesday. Ukip was nearly completely wiped off the map, gaining just one seat - from Labour - in Padiham and Burnley West.
The Copeland councillor, of Moresby Parks, will again face Conservative candidate Mrs Harrison in the general election, who served just two months in Parliament before the upcoming election was called by her party.
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A total of 4,851 council seats were up for grabs in 88 councils - all of those in Scotland and Wales and 34 in England - five weeks before the general election.
In Derbyshire, the Conservatives won 19 seats directly from the Labor, which would lose control of the council for the first time since the 1970s.
Its losses were centered in Lincolnshire, a former UKIP stronghold where Nuttall is standing for election on June 8, Hampshire and Essex, PA said.
A key plank of the Conservatives' general election strategy involves winning votes from Ukip and, on the evidence of Thursday's council elections, that's a winning formula. North of the border, the headline result is that Labour have lost overall control of Glasgow council for the first time in 40 years, with the SNP usurping them.
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The results sparked a furious reaction from the party's former donor Arron Banks, who said Ukip was "finished as an electoral force" under its current leadership.