Warning: amateur political science below
Stewart Wood has a well argued piece in the New Statesman, saying that it was the move by left and right towards a common centrism that laid the foundations for populism. Although parts of his argument ring true, I find others less convincing..Labour certainly moved to the centre and beyond in terms of its economic policies. The Conservatives moved to the centre in terms of social policy. But on economic policy, the Conservatives moved strongly to the right with austerity. Senior Labour thinkers still seem to have a blind spot on austerity.
Let me start by looking at the country that now has populism in spades as a result of electing Donald Trump as President. To argue that the Republican party has been moving to the centre over the last 30 years is obvious nonsense. The traditional centre right virtue of fiscal rectitude went out of the door with Ronald Reagan, and was completely ignored under the second Bush. The Republicans only extol the virtue of cutting budget deficits when they are not in power. When they are in power, they want cuts in taxes for the rich, increases in military spending but cuts in other government programmes.
Margaret Thatcher was considered pretty right wing when she was in power. Many of her key achievement in terms of her own agenda, such as a diminished union movement and shrinking the state through privatisation, were not reversed by Blair and Brown. It is difficult to argue that the Cameron/Osborne duo made any attempt to undo the Thatcher legacy. Instead they tried to go beyond it, by shrinking the state to a size relative to GDP not seen since the end of WWII. They did it under the pretense that they were forced to because otherwise the markets would no longer buy government debt. This was a colossal deceit. There no evidence that markets were concerned about government debt, and strong evidence that they were not.  This deceit should have become clear when Osborne cut taxes at the same time as continuing to cut spending.
Let me use a diagram to illustrate what I mean.  (The vertical axis could also be labelled 'identity' as well as 'culture'.) No doubt we could discuss the detail of the size and direction of the arrows, but I think this is roughly right.
In the US, the Republicans had moved steadily in a downward, socially conservative right wing direction, whereas the broad church that are the Democrats have largely remained in the same place (unless you go as far back as the southern Democrats). One possible argument is that this move by Republican politicians helped create the Tea Party. Republicans have always pretended their policies would help ordinary people, whereas in reality they have helped a rich elite. This has laid the ground for a populist leader who was prepared to move economic policy in certain respects away from the right (in particular advocating protectionism). The growing loss of respect of Republicans for their party elite allowed voters to ignore the views of senior Republican leaders in selecting Trump as their candidate.
In the UK David Cameron moved the Conservatives to become much more liberal, by in particular supporting gay marriage. (When I argued in an earlier post that the Conservatives had moved to the right, I was surprised how many comments I got back telling me this was nonsense, and naming gay marriage as the main reason why.) In the UK this left a large gap in this political space, which UKIP - the first successful mass party in England since the SDP - filled. As Jonathan Wheatley showed, UKIP members are social conservatives, but are much more left wing than the Conservatives in terms of economics.
Labour moved to the right under Blair, while remaining socially liberal. I agree with Stewart Wood that this alone was important in preparing the way for populism. As well as the lack of a major industrial policy, they did nothing to curb a rampant financial sector or reverse the gains of the 1% that were a feature of the Thatcher period, a point emphasised by Jean Pisani-Ferry in respect of both the UK and US. I think New Labour’s position is better described as liberal rather than neoliberal: New Labour substantially increased the amount of resources (as a proportion of GDP) going to the NHS, and they also did a great deal to try and reduce child poverty. Labour moved further right (and more neoliberal) as they became more accommodating towards austerity. It was hardly a surprise that party members tried to pull the party back by electing Corbyn as leader.
As I argued here, Brexit was a perfect storm where the economically left behind united with social conservatives. With Labour no longer seen as representing the working class, this allowed the right wing media (with the support of the Conservatives) to help convince the left behind that their problems were a consequence of immigration. The Leave campaign was populist in the sense I describe here: advocating a superficially attractive policy to some that would leave everyone worse off. Much the same is true for Trump, who won the electoral college by convincing the left behind that he really could bring back their traditional jobs, something he will be unable to do in any kind of general way.
So the idea of growing centrism in the US makes no sense, yet it is they who have just voted for a populist President. In the UK it only makes any kind of sense if you think in one dimensional terms.
 This is essentially because the Bank of England was pledged to buy whatever it took in the way of government debt to keep interest rates low.