The main problem with Pasokifying Labour is that there is no viable left wing party to replace them. As has been suggested, the best opportunity to create this would likely be in 1930s, with a socialist breakaway from a moderate Labour party that participates in a national government.
Other than that, I think our current era is the best opportunity to displace Labour with a more left wing alternative. It's difficult to create a strong socialist challenge in periods like the 1980s, because the left was in the ascendancy, and had no incentive to leave.
Its possible that the Lib Dems could position themselves as a centre left alternative to a centrist Labour Party, if Charlie takes the pledge, makes gains at the expense of Labour in 2010 and 2015 whilst the Tories somehow win and retain their majority or at least can operate a minority without the support of the LDs, Britain votes for Brexit, a more popular pro Brexit PM like Boris Johnson takes over, and the Lib Dems can place themselves at the head of anti-brexit, anti austerity sentiment against a Labour Party who has an indecisive leader that is trying to please all sides, like Burnham or Ed Miliband. To make thing worse, maybe you could spark a left wing resurgence within the party that means more infighting, but keeps the electoral college so that they the leadership itself remains in the hands of the moderates.
Or you could have a more classically socialist party emerge to challenge them. I am going to use this opportunity to plug a Respect wank idea for a TL I've been planning, where Ken Livingstone joins them rather than coming back to Labour in early 2004, and they become a viable fourth party, before leapfrogging the LDs in the following decade to become the third largest group in parliament, and create a left wing alternative to Labour. I hadn't planned to take Labour to under 20%, but I suppose that is doable in a situation where Britain votes remain, Farage remains as UKIP leader and the party surges, whilst a left wing alliance of a moderated Respect and the Greens run on an anti austerity platform, and the Tories corner the socially liberal middle englander vote under someone like Cameron or Osborne. There would be an awful lot of coincidental events involved though.
In any case, I'm not sure I can see Labour dropping below 20% purely due to the rise of a party to their left. There would need to be a more competent UKIP, or a pro Brexit Tory Party involved to threaten both sides of the Labour vote. Or maybe you could just make the UK an economic basketcase, not unlike Greece.
I definitely think the scenario of Burnham as leader, caught between an anti European populist right, and an anti austerity, pro European populist left, has potential.
The interesting thing about that comment is that is sort of what happened to the French Socialists. Corbyn turned out to have some campaign and political skills that Benoit Harmon didn't have. So you could do something like have a really incompetent version of Corbyn.
From what I gather, the main difference between Hamon and Corbyn was that the former had the image of a career politician, which allowed him to be easily outflanked by Le Pen to the right and Melenchon to the left, and Macron in the centre, whilst Corbyn had impeccable anti establishment credentials. So giving Labour a leader like a later Ed Miliband, Owen Smith, or Andy Burnham, who adopts radical rhetoric but come across as being a regular, triangulating politician, could do a great deal to weaken the party.