LONDON  - British Prime Minister Theresa May's Conservatives and the opposition Labour Party were both punished by English voters in a local election for the deadlock over Brexit, partial results showed on Friday.

With more than 40 percent of English local council vote results declared, the Conservative Party had lost 359 councillors and the Labour Party had lost 74 councillors, according to a BBC tally.

"It just seems voters, period, saying: 'A plague on both your houses'," said John Curtice, Britain’s leading polling expert.

"We knew that voters were unhappy with the way that the Conservatives had handled Brexit but looks as though they are also unhappy with Labour's response to the government's position on Brexit," he told the BBC.

As the two major parties, which are both nursing open fissures over Brexit, lost support, smaller parties gained.

The Liberal Democrats won 245 councillors, independent candidates won 188 council seats and Greens gained 34, the partial results showed.

British politicians and polling experts said the results showed that both major parties were being punished for the chaos surrounding Britain's delayed exit from the European Union and that they were losing their traditional dominance.

Nearly three years since the United Kingdom voted 52 percent to 48 percent to leave the European Union, there is still no agreement among British politicians about when, how or even if the divorce should take place.

Britain was due to have left the European Union on March 29, though May has been unable to get her divorce deal approved by parliament so she has turned to the Labour Party, led by socialist Jeremy Corbyn, in a bid to court his support.

It is still unclear how the deadlock might be broken, though there has been speculation that May might call a general election.

"The hold of the Conservative and Labour parties on the British electorate is frankly looking now as weak as it has done at any point in post-war British politics," Curtice said.

"Everybody who keeps on arguing that the only way out of the Brexit impasse is to have another general election should remember that there is perhaps a non-trivial probability that we would end up with another hung parliament," he said.

More than 8,000 seats on English councils - administrative bodies responsible for day-to-day decisions on local policy from education to waste management - are up for grabs. Full results are due later on Friday.

There are also some local elections taking place in Northern Ireland on Thursday but none in Wales and Scotland, which operate under a different schedule.

(Writing by Guy Faulconbridge Editing by Darren Schuettler, Robert Birsel) ((guy.faulconbridge@thomsonreuters.com; 07825218698;))