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London’s local elections battle has tightened dramatically with a week to go, leaving Theresa May facing at least one probable defeat, but potentially stopping Labour from taking two flagship Tory councils.

The final poll of the 32 borough elections suggests a fall in Labour’s lead in inner London that could cost Jeremy Corbyn victory in the high-profile targets of Wandsworth and Westminster.

The YouGov survey, commissioned by Queen Mary University of London, reveals the Conservatives trailing Labour by a huge 22-point gap.

But Labour is down three points after weeks of controversy over anti-Semitism in the party and Mr Corbyn’s judgments on the Salisbury attack, Russia and Syria.

Philip Cowley, professor of politics at QMUL, said the figures suggested “a good night for Labour”, which is on course to win dozens of seats, but that it will probably fall short of the two biggest prizes in its sights. Key findings suggest:

Barnet, where the Conservatives have been the biggest party since 1964, looks likely to go to Labour for the first time.

Wandsworth, Margaret Thatcher’s favourite council, looks set to remain under Conservative control but with a much smaller majority.

Westminster, which sets London’s lowest council tax, also looks likely to stay Conservative, along with Hillingdon, but both with smaller majorities.

Just 13 per cent of black and minority ethnic Londoners intend to vote Conservative next Thursday, while an overwhelming 75 per cent are backing Labour. The figures, which will horrify Tory modernisers, suggest the party’s name is toxic with black Londoners at a time of heightened sensitivity over the threats to deport Windrush migrants.

With seven days of campaigning to go, Labour is on 51 per cent (down from 54 in February), Conservatives are on 29 (up one point), Liberal Democrats unchanged on 11 and Greens on four, also unchanged.

In inner London, Labour is down eight points from February to 59 per cent, while the Conservatives are up five points to 22.

Conservatives still look vulnerable in Kingston and possibly Richmond, where Vince Cable’s Liberal Democrats are engaged in fierce local fighting, but the contests are too local for results to be inferred from a capital-wide poll.

Kensington & Chelsea is deemed unpredictable by experts after the Grenfell Tower fire.

Tory reprieves in Wandsworth and Westminster would reduce the danger of a leadership challenge to Mrs May which some MPs were linking with the scale of council defeats earlier this year. People who lived in Conservative-run councils were more likely to say their council was doing a good job — 48 per cent, compared with 43 per cent of people in Labour boroughs.

A quarter of Londoners felt their councils were “out of touch” or “wasteful”. Some 22 per cent said they were “bureaucratic” while 20 per cent described them as “incompetent”.

Some 18 per cent thought their council was “competent”, but just 11 per cent said they were “forward looking”, a tenth said “in touch” and just nine per cent called them “careful”.