Will you be voting for us?” asks the Liberal Democrat canvasser on the doorstep of an expensive-looking house in Putney. “No,” replies the middle-aged man standing in the doorway, “I’ll be voting for democracy.” We all understand what he means: Conservative, to fulfil the promise of the 2016 referendum and not simply to revoke Brexit, as the Lib Dems propose to do. My father, who lived most of his life just up the road from here, might well have given the same answer.

Putney, which at first glance looks like a very ordinary part of south-west London, is actually a cradle of English democracy. It was here, in the church of St Mary the Virgin, which still stands hard by the River Thames, that Oliver Cromwell’s soldiers gathered for a revolutionary debate in 1647. There the Leveller Thomas Rainsborough advanced the foundational idea of modern democracy in these immortal words: “I think that the poorest he that is in England hath a life to live as the greatest he, and … every man that is to live under a government ought first by his own consent to put himself under that government.”

Centuries later, the constituency of Putney could be a bellwether in the UK’s latest, hugely consequential democratic choice. If Putney remains in Conservative hands, then Boris Johnson will probably secure a clear majority in parliament and take Britain out of the EU on 31 January. If it goes back to Labour, then there is still a chance of a multiparty parliamentary majority for a second referendum, in which the country could democratically choose to remain in the EU. My father might have agreed with the man on the expensive doorstep but I emphatically don’t. As the leading Brexiteer David Davis once put it, if a democracy cannot change its mind, it ceases to be a democracy.

To achieve that result will require skilful tactical voting on a large scale, here as in other marginal constituencies. In the 2016 referendum, Putney voted 72% for remain. Its Victorian terraced houses, comfortable Edwardian semis and more modern mansions are full of Tory remainers. The local electoral history and recent in-depth polling by the campaigning organisation Best for Britain clearly indicate that, if Brexit is still the key issue for these Tory remainers, they need to hold their noses and vote Labour. So should the constituency’s large minority of Liberal Democrats.

Elsewhere, for example in the Oxford West and Abingdon constituency where I now live, Labour voters should return the compliment and vote Liberal Democrat. According to Best for Britain, adept tactical voting by fewer than 41,000 voters in 36 marginal constituencies could prevent a Conservative parliamentary majority. There is now a large degree of consensus between the different tactical voting advice sites. Everyone who believes this election is our last chance to stop Brexit should simply follow that advice, whatever you think of the favoured party or candidate.

Having grown up in the urban village of Roehampton, which is part of the Putney constituency, I find it a curiously moving experience to be back, observing political canvassing amid the characteristic greeny-yellowy-grey brick terraced houses and ornate Victorian pubs. My parents moved here because my mother’s uncle was the local Conservative MP. Uncle Hugh, as we knew him, was elected in a byelection in 1942 and held the seat until 1964, when the tower blocks of the large new council estate built in Roehampton swung it to Labour. It has gone to and fro between Tory and Labour ever since, most recently being held by the popular Justine Greening, one of the pro-European Tories to be stripped of the Conservative whip by Johnson, and as a result now retired from politics. (If the Conservative leadership had kept her in a broad church party, she would surely have held this seat for them.)

When I was a child, my staunchly Conservative father would sometimes take me with him when canvassing for the Tories in those tower blocks. I will never forget peering up from the height of Dad’s thigh to see the same scene repeated all the way from the first to the 10th floor: council flat door opens, man or woman with cigarette looms above me and gives my father some more or less polite variant of “I’m not voting for you, mate.” Now I’m standing at some of those same doors, in those same blocks: nothing has changed and everything has changed.

My rain-drenched return to Putney leaves me depressed and heartened in equal measure. Depressed because here, as elsewhere, the disastrous disunity in the pro-European camp could hand victory to Johnson and the Brexiteers. The People’s Vote campaign, which magnificently brought a million people to demonstrate on the streets of London, has collapsed in internecine feuding. Labour and Liberal Democrat candidates are not stepping down to help each other where they clearly should. The yellow wave of liberal revival has not materialised, partly due to two major blunders by rookie Lib Dem leader Jo Swinson: proposing to revoke article 50 without a second referendum (democratically illegitimate in principle, tactically foolish in practice); and placing herself front and centre of the campaign as (pull the other one!) our next prime minister.

With both Labour and Liberal Democrats campaigning hard in Putney, a sophisticated YouGov poll currently predicts that they will split the opposition vote, thus gifting a narrow victory to the Conservatives. With Britain’s unfair first-past-the-post system, all the dedicated efforts of the Lib Dem and Labour canvassers I watched pounding those rainy pavements would then count for nothing.

I’m also depressed by the fact that these tower blocks are exactly as I remember them, only much more run-down, while just a stone’s throw away countless millions have been spent on doing up the private houses of the prosperous. “The poorest he that is in England” couldn’t afford even a broom cupboard in Putney these days. When I quote an online estimate that the average house price here is more than £500,000, several local people have the same response: “Oh, that seems a little low to me.” Meanwhile, the food banks are called upon more than ever and the curate of St Mary’s tells me, “We had 35 homeless people sleeping in the church on Monday.”

But I’m also heartened. Heartened because in the very place where the English reinvented democracy, democracy is alive and flourishing. I spoke to the three main candidates and found them all to be decent, civilised, serious people. Everywhere I went I saw little groups of canvassers busily going from door to door, typing their findings into their smartphones (“Would you call her soft Conservative?”). While the candidates have faced isolated incidents of abuse, for the most part the people of Putney uphold the best English traditions of robust civility. The voters on the doorstep were also impressively thoughtful, and many of them already display a laser focus on tactical voting. So this unlikely cradle of democracy might yet help keep England where it belongs: in Europe.

• Timothy Garton Ash is a Guardian columnist