Sadiq Khan is London's first Muslim mayor

London became the first major Western capital to elect a Muslim mayor on Friday as Labour claimed victory for its candidate Sadiq Khan despite setbacks elsewhere in Britain for the main national opposition party.

Labour leader Jeremy Corbyn hailed Khan's win over Conservative Zac Goldsmith, after a bitter campaign in which Prime Minister David Cameron sought to link Khan to Islamic extremists.

"Congratulations Sadiq Khan. Can't wait to work with you to create a London that is fair for all!" Corbyn wrote on Twitter ahead of the publication of final results.

The race to replace the charismatic Conservative Boris Johnson pitched Khan, the son of a bus driver and a seamstress who emigrated from Pakistan in the 1960s, against Goldsmith, whose father was a wealthy tycoon.

Khan's victory offered some cheer for Corbyn after Labour's performance in elections across Britain in the wake of a row over anti-Semitism in the party failed to dispel questions over the veteran socialist's leadership.

New York Mayor Bill de Blasio was among the first to tweet his congratulations to "fellow affordable housing advocate, @SadiqKhan. Look forward to working together!"

Paris Mayor Anne Hidalgo added on Twitter that Khan's "humanity, progressivism will benefit Londoners."

There was no immediate reaction from Goldsmith, although his sister Jemima, the ex-wife of Pakistani cricketer and politician Imran Khan, said Khan's victory was a "great example to young Muslims".

In reference to the negative tone of the race, she said her brother's campaign "did not reflect who I know him to be -- an eco friendly, independent-minded politician with integrity".

The victory was also hailed in Tooting, a multi-ethnic area of south London where Khan lives.

"Sadiq Khan will have a unifying factor because he is Muslim, an immigrant, he is from working class, so he understands the working class people and he can associate with them," said Shahzad Saddiqui, a local businessman.

Labour losses elsewhere

Khan's party fared less well in other local and regional elections on Britain's "Super Thursday", in which 45 million Britons were eligible to vote.

For first time in decades, Labour came third in elections for the devolved government in Scotland, behind the Conservatives, in a vote won by the incumbent pro-independence Scottish National Party (SNP).

The party retained its grip on power in the Welsh assembly, although it lost one seat.

But it failed to impress in local elections in England. With results in from 115 out of 124 councils, Labour had control of 57, down one, and 1,265 seats, down 19.

Cameron's Conservatives had control of an unchanged 31 councils and 708 seats, down 18.

Corbyn, who has faced opposition from centrists in his party since being elected last year, insisted his party had "hung on" and surpassed expectations.

But critics point to the long tradition of opposition parties benefiting from mid-term elections, and noted Labour had bucked that trend.

Matthew Goodwin, politics professor at Kent University, said the party was in "serious trouble".

"The Labour party is now third in Scotland for the first time since 1910 and it has failed to make a serious impression in southern England," he said.

'Persuade not divide'

SNP leader and Scottish First Minister Nicola Sturgeon will lead the separatist party into its third successive government in Edinburgh, although the party lost its outright majority.

She announced she would lead a minority administration, and played down talk of a fresh independence referendum to follow the unsuccessful one in 2014.

"The SNP will always make our case with passion, with patience and with respect but our aim is to persuade not to divide," Sturgeon said.

The other big story in Scotland was the success of the Scottish Conservatives, who came second with 31 seats.

The party has been deeply unpopular in Scotland since the 1980s premiership of Margaret Thatcher but its fortunes have turned around under current Scottish leader Ruth Davidson.

Davidson is a charismatic and openly gay 37-year-old whose cheery, no-nonsense style and proficient use of social media has fuelled her party's success.

Cameron, who is grappling with deep splits within his party ahead of the June 23 referendum on Britain's membership of the European Union, said the party's showing across the elections was "remarkable".