With four days to go until polling, Labour volunteers are busily unloading huge boxes of campaign leaflets at their overcrowded election HQ in Battersea, south London. As fast as the latest material arrives in vans from the printers, it disappears out of the door again, as canvassers grab handfuls and fan out across the streets of Wandsworth.

This is the most important, closest and most keenly fought local election battle here in decades and canvass returns suggest the result is poised on a knife edge. “I would say the chances are 50/50,” says Simon Hogg, the Labour leader on Wandsworth council, which has been held by the Conservatives since 1974.

For Labour to storm this Tory fortress – Wandsworth was Margaret Thatcher’s favourite council and it boasts the country’s lowest council tax – would be a massive story. “It would be a political earthquake,” Hogg says. “And I think we can do it. More and more people are coming over to us in wards we have not won since the 1960s and 70s.”

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The aftershocks would be felt nationally as well as locally. With Theresa May under pressure over immigration and the Windrush scandal, and struggling to prevent her party splitting irreparably over Brexit, a Labour win in Wandsworth and other urban Tory heartlands would ask still more questions about the prime minister’s chances of long-term survival in No 10.

Out on the streets, Conservative-to-Labour converts are not difficult to find. A third of residents in this part of south London live in rented properties and the lack of affordable housing is a huge issue not just for tenants but for those who own their properties and those who worry their children will never be able to afford to leave home. Brexit, too, is on many voters minds, with Remainers outnumbering Leavers by a large margin.

As Labour activists turn into her street, Sheila Arnold, 71, comes to the door. She has voted Tory all her life, as did her father before her. But she has had enough. “I’m going Labour this time and I never have before. There are not enough houses for the young people round here and that is not right.”

Other things have upset her about the Conservatives. She agrees with them that immigration needs to be controlled but when she heard how May’s government had treated the Windrush generation she was horrified. “I think it is awful. We do have to control immigration but these people were invited in by us, and they helped us. This is their country now, their home, and they have been treated like muck. It is really terrible.”

Across the road Remi Tangian, a domestic helper who came to the UK from the Philippines 30 years ago, feels the same way about how the Conservatives have treated the Windrush generation and is switching her vote. “It is so unfair,” she says. “We have to have change. I cannot agree with what they have done.”

We are aware that we would be making history if we were to win Adam Hug, Labour, Westminster council

Time on the campaign trail suggests the wind is driving hard into Labour’s sails ahead of polling day. Yet there is a nervousness. Fear of falling just short is competing in the minds of party activists with excitement at the prospect of winning. What could be a great night in Wandsworth and other urban areas could just as easily turn into one of deflation and dejection.

An opinion poll on Friday suggested Labour would gain a large number of seats on Thursday – but still fall short of taking the two great London prizes of Wandsworth and Westminster. As one Labour activist out on the streets put it on Friday afternoon: “The challenge for us is leave no stone unturned and whip up the sense we can do it, without raising expectations so high that we shoot ourselves in the foot if we fall short.”

Across the capital, it is same story in traditionally Tory Westminster. The council has never fallen to Labour before. Currently the Conservatives hold 44 seats to Labour’s 16, meaning Labour needs an even bigger swing than in Wandsworth.

Concerns about Brexit among its largely Remain electorate, plus worries about a lack of affordable housing and the memory of the Grenfell Tower disaster across the border in Kensington and Chelsea, all mean this election could go any way. That Adam Hug, the council’s Labour leader felt it worth his while to tramp through the borough on a wet Friday was evidence enough that he believes he has a chance. “We are aware that we would be making history if we were to win,” he says.

Labour leader Jeremy Corbyn on the campaign trail in Grimsby. Photograph: Danny Lawson/PA

London seems certain to provide the big stories on Thursday: 40% of the seats up for election are in the capital. If Labour fails in Wandsworth and Westminster it is more likely to succeed in Barnet, despite worries its vote could be negatively affected in its large Jewish population by the party’s ongoing antisemitism rows. Labour also has its eyes on Trafford, the Tory-held area of Greater Manchester.

Across the country more than 4,300 seats are being contested in 151 mainly urban councils. Analysis by Colin Rawlings and Michael Thrasher at Nuffield College, Oxford suggests that the Conservatives will lose 75 seats, with Labour gaining 200 and the Lib Dems 30. Ukip could lose most, if not all, of the 163 seats it gained last time. As Rob Ford points out on page 10, where former Ukip voters put their cross on Thursday will be one of the key psephological questions of election night.

The vote will also be a key test for the shrunken Liberal Democrats and their leader Vince Cable. The party’s strength was built from strong roots put down over decades in local government. Occupying a distinct position on Brexit, as the only mainstream party implacably opposed, its supporters hope it will be able to mop up votes in Remain areas.

The party has been pouring its resources into trying to retake control of Richmond and Kingston upon Thames in London from the Tories, and if it can it will have a decent story to tell on election night. Failure to do so – coupled with an inability to defend the London borough of Sutton – would raise further questions about what the party is for these days.

For all the major parties the verdict of the electorate on Thursday and their reactions to the results on Friday will send important signals.

Will Labour and Jeremy Corbyn be able to convey a sense of momentum sufficient to suggest the party is on course to win back power at national level? For the Tories, will May and her party be able stave off symbolic defeats at the hands of Labour in London and elsewhere, and steady their nerves?

Can the Lib Dems show they can again be a big force in local government and begin rebuilding? Can Ukip simply survive?