The buzz was not allowed to last even a day. Less than 12 hours after Barack Obama had wrapped up his party's convention in Charlotte, the latest version of the number that could determine the fate of his presidency was published: unemployment was down slightly, to 8.1%, but the new jobs total was disappointingly below expectations.

If that killed the Democrats' mood, Republicans could be forgiven for wondering if their convention last week in Tampa ever happened. The race appears to be pretty much where it was before, the latest poll of polls showing Obama and Mitt Romney deadlocked on exactly 46.7% each, even if in the key states the president retains the advantage.

So what will be left behind from this quadrennial ritual, a fortnight of back-to-back gatherings of America's two main political parties? We learned anew the curious paradox of Obama: that the man hailed as an orator of Ciceronian power four years ago is a poor communicator, at least when it comes to his own record. He has notched up significant achievements – averting a second depression, reviving a dying auto industry, passing the healthcare reform that eluded a century of predecessors, not to mention (though he does) Osama bin Laden – and yet he cannot seem to turn that scoresheet into a compelling narrative. His speech on Thursday disappointed even sympathetic listeners. Perhaps stung by the accusation that he offers flowery talk instead of action, he strove too hard to speak practically rather than soar.

The best advocate for the Obama presidency turned out not to be Obama but Bill Clinton, who gave a masterclass in how to dive into substantive, even nerdy, detail and still connect emotionally. It may be 16 years since he last raced competitively, but Clinton remains a world champion of politics.

We learned too that the spouse's speech is now a central part of any presidential campaign: woe betide the candidate whose partner cannot address a full arena and TV audience of millions. Both Ann Romney and Michelle Obama did better than their husbands. If the latter chooses to follow the Hillary precedent, and a senate vacancy opens up in Illinois, she would walk it. She's a natural.

All of this was watched by large numbers of Americans, of course. Incredibly, more tuned in to see Bill Clinton, a former president delivering a political speech, than to watch the season-opening Giants v Cowboys ballgame. But also paying attention were several British politicians. Tories among the Republicans included Sayeeda Warsi, Daniel Hannan MEP and the irrepressible Steve Hilton – along with, more surprisingly, Ed Miliband's righthand man, Stewart Wood. Wood went on to Charlotte where he was joined by David Miliband and Douglas Alexander. Flying the Tory flag among the Democrats was the American-born MP Tobias Ellwood.

Given that such a large British contingent is probably not found at, say, a French or German party conference, it's worth asking what these UK politicos expected to learn. Take the Conservatives in Tampa. They might think of the Tories and Republicans as sister parties, but the gap between them is now surely so wide as to render most of what they saw inapplicable.

Republicans express a loathing for the very idea of government more extreme, or at least more explicit, than anything you'd hear from a Tory. Cameron's Conservatives at least feel obliged to pay homage to the NHS, an idea condemned as socialist tyranny by any Republican who's heard of it. Cameroonian modernisation might seem a distant memory after this week's rightward reshuffle, but David Cameron's liberal stands on equal marriage or abortion ensure he would not survive a week in today's Republican party.

So the transatlantic trade in political ideas is always going to be bumpy. Still, there are some items I assume those returning pols have stashed into their hand luggage. For Labour, item one is surely a DVD of the Clinton speech: Ed Miliband should sit down, pen in hand, right away to watch and learn.

He would conclude, first, that a politician does not have to talk down to an audience. It is possible to talk seriously. Indeed, if you show the voters you respect them, they'll respect you. Second, it's wise to deal with the opposition's arguments, rather than hoping they'll go away. Clinton went through the Romney-Ryan chargesheet and tore it apart. Labour must do the same with the persistent claim that it cannot be trusted to run the economy because it overspent last time. It takes effort, but it's worth it.

What's more, Clinton showed the power of arithmetic. He walked through the Republicans' numbers, exposing that their sums did not add up: you cannot cut taxes, spend more on defence and cut the deficit. Labour has to persist making the apparently counterintuitive case that austerity in a recession actually adds to, not reduces, the country's debts because it kills growth.

Third, framing is all. Clinton is uniquely skilled at formulating an election in the terms that suit him and his party, all the while sounding fair and reasonable. This is not about phrase-making. The analysis comes first; the language follows. On Wednesday the former president framed the coming contest not as a referendum on Obama – which Obama would lose – nor as a question of whether Americans are better off now than four years ago, but as a choice between a "winner-take-all society" and one of "shared prosperity and shared responsibility".

Labour has something to learn from the Republican convention too, but it's uncomfortable. Romney's troubles show that, even when faced with an incumbent mired in dreadful economic numbers, a challenger will struggle if he is seen to lack personal appeal. British politics is not yet presidential, but it has been heading in that direction for decades. Whatever other messages Labour projects, the voters have to see a party leader as a potential prime minister. That is a basic threshold to clear.

But this cuts both ways. Ellwood and the others should report back to Cameron that if even the well-liked and ultra-charismatic Obama cannot surge ahead of an opponent as weak as Romney, it proves just how hard it is for incumbents to get re-elected in a recession. (Just ask Nicolas Sarkozy.) Candour should compel them to add that he cannot keep on doing things that alienate women – witness this week's demotion of female ministers. The Republicans, still reeling from Todd Akin's talk of "legitimate rape", have made that mistake repeatedly and are paying for it with a wide gender gap.

Of course the two countries are different. But some things in politics are universal. In November we'll see the proof.

Twitter: @j_freedland