Britain doesn't usually count for much in the Middle East, but this time it could make all the difference. As the Palestinians seek United Nations recognition as a state, a quirk of diplomatic algebra leaves Britain with a chance to play the decisive role – and to complete some unfinished business dating back more than 60 years.

Barack Obama has already said the US will vote against any Palestinian move towards statehood at the UN general assembly now gathering in New York. Large swaths of Latin America, Africa, Asia and the Middle East plan to vote for it. Which leaves Europe as the diplomatic battleground. If the leading European powers side with the US, the Palestinian initiative will be seen as a failure. If an EU majority backs recognition in some form, the Palestinians can claim symbolic victory.

Already negotiations are under way, both among the European nations and between the EU and the Palestinians, aimed at reaching a common, compromise position. France and Spain want to say yes, Germany and Italy are wary. Which leaves Britain with something akin to a casting vote in the "quintet" of leading European nations. How David Cameron jumps will be crucial in determining Europe's stance, and therefore the fate of the Palestinian effort itself. For decades Britain has talked about punching above its weight. Now its weight really counts.

The backroom haggling concentrates on which UN body will make the decision – the general assembly or the security council – and what exactly they'll be voting on. If the Palestinians aim high, they'll apply to the security council for full UN membership, where Obama has promised they will be greeted by a US veto. Or they could go before the general assembly, where 140-odd countries are ready to grant the lesser prize of an upgrade in UN status, from observer to "non-member state", with access to some of the major international institutions. Devil's in the details and all that, but Britain's attitude should be clear: we should say yes.

That's because UN recognition of a Palestinian state in the territories occupied by Israel in 1967 will breathe fresh life into the ailing idea which, despite everything, remains the last best hope of Israeli-Palestinian peace – a two-state solution. By recognising a state of Palestine alongside Israel, the UN will entrench the notion that the only way to resolve this most stubborn of conflicts is for these two nations to divide the land between them into two states. In so doing it will halt the steady drift, born of despair more than enthusiasm, towards the so-called one-state solution – so-called because while it would bring one state, it offers no solution, just a single entity that would frustrate the yearning for self-determination of both sides.

The two-state solution has been on life support for years now, its health deteriorating since Binyamin Netanyahu returned to the prime minister's office. Officially he subscribes to two states, yet his every policy action, typified by unceasing settlement building in the West Bank, puts that goal further out of reach. A loud yes vote at the UN would reverse that trend, renewing what has long been the global consensus: that the land of historic Palestine has to be shared between the two peoples who live there.

Here's where Britain and Europe can give a little extra help. A new and insightful policy document by the European Council on Foreign Relations – titled Why Europeans Should Vote Yes – suggests the new UN resolution could explicitly support the idea of "Israel alongside a Palestinian state, thereby entrenching Israel's legitimacy and its permanence". Having the general assembly, including its Arab and Muslim member states, vote for such a resolution would amount to de facto recognition of Israel – and reassure those who fear the country's "delegitimisation". The text might even reconfirm UN resolution 181, the original 1947 partition of Palestine into two states, one Jewish and one Arab. Renewing 181 would complete two items of unfinished business. First, that Palestinian state promised 64 years ago never materialised: its land was gobbled up, the West Bank taken by Jordan, Gaza by Egypt and much of the rest by Israel. A yes vote next week would finally acknowledge the Palestinian right to lands they were meant to govern decades ago. Second, Britain abstained in 1947; now it has a chance to say yes to the partition of the land it once ruled.

Still, it's the future we should be imagining, specifically the day after a US- and Europe-led no vote. Palestinian public opinion would surely conclude that the path of nonviolence and diplomacy had failed, shunned by the very countries who had repeatedly urged them to take it. In the ongoing argument within Palestinian society, the advocates of armed resistance would appear vindicated, their opponents humiliated.

Imagine the contrasting scene in Israel, where Netanyahu would be doing a victory dance. As Daniel Levy, co-author of that ECFR paper, told me, a European no vote would reward the Israeli PM's stubbornness: "He will respect the EU even less, and it would entrench his rejectionism even more." Bibi would taunt those who had warned of a September diplomatic tsunami as "liberal crybabies", unable to see that tough intransigence always wins the day. A prime minister who should be on the ropes – assailed for watching as two former allies, Egypt and Turkey, make common cause against Israel – would instead be hailed as a maestro of international power politics.

If the prospect of boosting Bibi and discrediting Fatah does not deter European governments contemplating a no vote, perhaps they should think on their reputations in the region if the Palestinians are thwarted. Having praised those peoples who seized their own destiny through the Arab revolutions, they would have denied, however symbolically, that same path to the Palestinians. Obama is already fated to be condemned as a hypocrite by the Arab world, thanks to his promised veto. If the Europeans make the same mistake, they will lose whatever influence they retain in the Middle East. No one will listen to a word they say.

There are misgivings among Palestinians and their supporters, of course. Some worry that recognition of the Palestinian Authority would diminish the PLO, which represents the wider Palestinian diaspora. The glib answer is that the Palestinians of the occupied territories have been dominant since at least the Oslo accords, signed 18 years ago today, and that a UN vote will only formalise what is already true. More subtly, such a usurping of the PLO would only be in prospect if the Palestinians started implementing practical statehood, declaring interim borders on the West Bank and the like. And no one believes that is likely.

The truth is that, by itself, a positive UN vote will not change the lives of too many Palestinians. But a negative response would be a disaster, boosting Israeli hardliners, weakening Palestinian peacemakers and choking the near-dead two-state solution. All three of those arguments should resonate in European capitals, but the last two should hit home in Israel itself. That is why a wise Britain would vote yes at the UN – and why a shrewd Israeli government, one that understood the best form of security is peace, would vote exactly the same way.