Even before David Cameron had opened his mouth, a Scottish friend of mine got in touch. "Today reminds me of an old joke," his email began, "about a pilot announcing that he'll have to make an emergency landing in the sea. Panicked passengers ask the flight attendant where the life vests are. 'Oh,' says the flight attendant, 'so now you're interested.'"

For Scots, especially those who have grappled with the question of home rule for 35 years or more, the British prime minister and the rest of the London political class have left it awfully late to start paying attention now. With just seven months to go till Scots vote yes or no to independence, the rest of the United Kingdom is like a husband whose wife has been threatening divorce for three decades – but waits till she's got a suitcase in her hand and her coat on before looking up from the couch to say: "Can't we talk about this?"

Cameron mentioned divorce in the speech – more a plea, really – that he delivered in the Olympic Park in London today, but this is not only about him. It's not even about the Westminster village, though it would be nice to dump all the blame there. We could throw in the media while we're at it (even the Guardian would not be exempt). But the truth is, the whole of non-Scottish Britain is implicated in this one. Wherever you look, the attention paid to Scotland's choice has been scant. And that's putting it nicely.

Cameron was right to say that the response of the country that could become rUK – the rump United Kingdom that will be left behind – on 19 September has consisted either of a two-fingered good riddance; a regretful sigh, resigned that there is nothing to be done; or else a neutral, unbothered shrug of the shoulders. The PM was surely right too that the first reaction – outright hostility – is rare. Lethargy and inaction dominate.

It's odd, this reaction. If France or the US were facing the possibility of losing a third of their landmass and a tenth of their people, you can bet they'd be at least curious. But here the apathy is deafening.

Some will say that's only because the prospect is not serious, given that polls show the no side comfortably ahead. If that's the explanation, it could be resting on a faulty premise. As the SNP leader, Alex Salmond, likes to point out, the last seven polls have tightened – in favour of yes.

More to the point, and as a series of conversations in Edinburgh and Glasgow this week demonstrated to me, previously hard ground is shifting in Scotland. Once solid unionists, including those from staunch Labour backgrounds, are at least considering voting for independence. Even activists for the save-the-union Better Together campaign admit that where there is movement, it is from no to yes: next to no one is moving in the other direction. Unionists also admit that in Salmond's SNP they face a rival that can summon deeper resources, more activists and greater enthusiasm, as well as a formidable on-the-ground operation.

But the shrug shown to Scotland by the rest of Britain is about more than a reading of the polls. There has also been a failure – and I include myself in this – to take seriously the motivation for independence. If London-based commentators like me have engaged in this debate at all, it's mainly been to make the case for the union and leave it at that. Too few of us have probed deeply into why so many Scots want to break away.

It's not about blue-faced loathing of Sassenachs. It's not about an easily patronised desire for passports, border fences and anthems. In fact, plenty of yes voters I spoke to understand that some of the traditional trappings of statehood won't be theirs anyway. The plan is to keep the pound and, as the governor of the Bank of England explained last week, that will mean "some ceding of national sovereignty", even a version of fiscal and political union. So this will not be independence in the classic, 19th century sense.

What's driving so many Scots to consider saying yes has less to do with their view of Scotland than what they believe has happened to Britain. Again and again, from people who would never describe themselves as nationalists, I heard the same story. Since 1979 Britain has been breaking away from what used to be called the postwar settlement. Led by an overdominant London and south-east, British politics has been tugged rightward. The prevailing ethos of the past 35 years has been one of turbo-capitalism, privatisation and a shrinking welfare state. Yes, the process was begun by Margaret Thatcher, but Tony Blair and Gordon Brown did little to stop it, and in some cases accelerated it. And Scotland wants no part of it.

Some have interpreted that as a claim to Scottish moral superiority. When we met in his Edinburgh front room, Alistair Darling, head of Better Together, condemned such "arrogance" along with the notion that somehow "the Scots are inherently better people". Perhaps some do conceal that belief in their talk of a distinct, Nordic-style culture of social democracy in Scotland. But the facts are the facts. And the facts say that, while UK politics can seem like a Dutch auction between the Conservatives and Ukip over who can bash Europe, repel immigrants or slash welfare with most vigour, Scottish politics is usually a competition for the terrain of the centre-left. This week the two main parties, Labour and the SNP, jostled over who has the best plan to scrap the bedroom tax. Meanwhile the SNP has a striking policy on immigrants: it wants more of them.

This, then, is what's driving so many Scots to consider making the break: a despairing fear that, given the way a few marginal seats in middle England can decide UK elections, Britain will never again return to the kind of social democratic values that still find a ready consensus in Scotland. It's not that the Scots are leaving Britain – it's that Britain has left them.

Viewed like this, the campaign for independence requires a different response from those outside Scotland. It is devilishly difficult, as Cameron's speech showed. That he delivered it in London highlighted the most obvious problem: that for the English to get involved immediately looks like hectoring from on high. When the case for the union is made by a southern English Tory public schoolboy, it simply reinforces everything yes voters want to get away from.

So perhaps the best approach for non-Scots minded to follow Cameron's advice, and phone a friend or relative north of the border, is to focus not on what the union should mean for them but what it does mean for us – to set out what we, not they, would lose if they voted yes.

The loss for Britain's progressives would be great indeed. Gone would be that tug leftward, that counterbalance to the politics of the overheated English south-east. This is not abstract. If Scotland goes, so too will 59 MPs for Scottish seats – only one of whom is currently a Conservative: the bloc that has made Labour governments possible. "No more Tory governments. Ever" is a current yes slogan. But a yes vote could mean the reverse for rUK: "Permanent Tory government. For ever." That will matter for an independent Scotland, especially one bound to this eternally Tory Westminster by a fiscal union. An independent Scotland might have less scope to be social democratic than it does now.

There will be another loss, too. "British" has become a capacious, even a generous, category. It hyphenates easily. Because it always stood for a plural, multinational identity, it is able, by definition, to accommodate difference: you could always be Welsh-British or Scottish-British, so now you can be Black British or Muslim British.

"Welsh" or "English" have not functioned the same way. That's not to say they can't. If the Scots leave the union, those left behind will have to make, say, "English" a looser, more inclusive category. But that is the work of at least a generation. And it will feel like building from scratch a house we built long ago.

So I no longer dismiss the Scottish yearning behind yes. If I were a Scot, I might well be leaning that way myself. But since I'm not, I can only plead with them to stay. I can see what they might gain by leaving. But it will be our loss.

Twitter: @Freedland