As you drive south from Derry and turn right to cross the Foyle river by the Asda outside Strabane, nothing tells you that you are about to leave one country and enter another. There are no signs, no flags, no police and, though I may have missed them, no security cameras either. It’s only when you see the words Bus Éireann on the stop on the far side of the bridge that you get any inkling that you have left the United Kingdom and are now in the Irish Republic.

A couple of weeks ago I took a road trip around the British Isles. We went from London up to southwest Scotland, then on through Northern Ireland – via Derry and Strabane – to County Sligo, before returning via Dublin and Holyhead and home through north Wales. We took in five countries in six days, crossed five borders, and I never once had to show my passport at any of them. I hadn’t expected to do so. But for how much longer will this be so?

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Though the countries of these islands are very different in many ways, and all are assertive about their identities, we are all used to the boundaries between them being more cultural than political. Only Scotland had even erected a road sign that announced that here was a different place from the one we had just been. The others simply merged into one another. I increasingly came to suspect that this is because, in spite of their differences and their histories, all five countries of these islands still have at least as much in common as they do that separates and divides them.

Will this survive? The United Kingdom is not as united as it was in my boyhood. Scotland hovers constantly on the threshold of separatism. Brexit has thrown Irish politics into uncertainty. Labour-majority Wales chafes at dependence on the Tory government in London. Meanwhile, England has pulled up the mental drawbridge against Europe without giving a moment’s thought to the dire effects on Ireland, Scotland or Wales. Yet it remains true that we are bound together in deep and extraordinary ways.

One is the language. Travelling to WB Yeats country on the Irish west coast, reading Roy Foster’s biography of the poet as I did, then stopping at the National Famine Museum at Strokestown, the Anglo-Irish dimension loomed very large. But I have never been more conscious, in Ireland and in Wales, of how pathetically little effort English people like me make to understand or speak a few words of the local languages. An ice-cream seller at the foot of Snowdon said to me once: “If you were in Italy, you’d try to ask me in Italian, so in Wales why don’t you try to ask in Welsh?” He was right.

Yet the Irish Republic really is a different country. Men and women have fought, died and suffered for it to be so. Any suggestion from an Englishman that we share a single geographic or social space with Ireland can still provoke accusations of colonialist sympathies. I once wrote a Guardian editorial about the BBC weather forecast in which I gently suggested that Dublin might perhaps be usefully identified on the weather map. In response, some letter-writers made it very clear that, after a thousand years of oppression, Ireland’s weather was none of the Guardian’s damned business.

Facebook Twitter Pinterest Constance Markiewicz, the first elected woman MP in the UK. Photograph: Walshe/Getty Images

But these islands are our shared business, or they should be. In Sligo, as the Yeats summer school gathered next door to discuss Yeats and Asia, the woman in the tourist office asked me: “Do you know who Constance Markiewicz was?” I certainly did, but it was striking that she asked. The woman born Constance Gore-Booth in a landed Anglo-Irish family in 1868, whose home was up the road from Sligo, would become Britain’s first elected female MP (though as a Sinn Féiner she never took her seat), and later, Ireland’s – and the world’s – first female cabinet minister.

Does Theresa May know who Markiewicz was? I have no idea. May was in Belfast the day before I arrived there. She said, as she always does, how precious the union with the north is to her. Yet her view of the union is as rigid as a flagpole. It is a home counties English Conservative view that has little feel for the human and cultural suppleness of the UK union, or of the European one either. She gives no inkling that she possesses the emotional or imaginative space for the complexity of these islands or their history. You would never know from her Belfast speech that Northern Ireland voted 56%-44% against Brexit, let alone that support for remain in Northern Ireland has risen to over 70% in a recent poll – still less that her government is propped up by the DUP, which she never mentioned.

Do not mistake shared institutions for shared and supple mentalities. That was one lesson I learned on this drive. In Kirkcudbright, a nice town in Scotland, it was Scottish night as we arrived, and a bagpipe band straight out of a tourist brochure marched through the town as dusk fell. But the children who paraded with flags before the pipes arrived had Union Jacks as well as saltires, along with all the flags of Europe and beyond. The following evening, hundreds of people on horses “rode the marches” – a form of horseback pub crawl to commemorate the town’s burgh status – but all wearing the countryside riding kit you would see in Wiltshire.

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Returning across the Irish Sea to Holyhead, we detoured south to Llanystumdwy in north Wales to visit the grave and museum dedicated to David Lloyd George. Britain has never had a prime minister who more confidently combined the particular with the shared identities on which these islands rest, nor one who believed so firmly, or so early, in the devolutionary “home rule all round” approach that the Liberals came so close to embracing in 1912. Nor one who spoke two British languages fluently and with such brilliance. Nor one who had more feel for the islands’ personalities. And yet, Lloyd George’s feel never extended to Ireland, which he only visited once and of whose modern divisions he is one of the prime authors.

At the risk of employing national stereotypes, all the UK countries have much to learn from the Irish Republic, which seems in many ways to be more at ease with itself than Scotland, Wales, Northern Ireland and England – England above all. If even Lloyd George, with his enormous leadership talents, failed to keep these islands together at a time of crisis, when Ireland revolted during and after the first world war, it is difficult to imagine that May, who is much less gifted, can do so. All I know is that the solution, if there is one, lies in mutual respect, historical humility, and an open and curious mind about the extraordinary part of Europe we all inhabit.

• Martin Kettle is a Guardian columnist