It's hard to say how much the alchemy has changed but in early 2016 the idea of leaving the European Union was unthinkable to Parliament - which is why a referendum was promised in the first place. So I suspect, in the absence of party politics, there would be a majority for a second referendum.
Labour's position on Brexit has been farcical and they've aided and abetted the Tories for the past two and a half years. It's fair to say that the senior Labour leadership are entirely satisfied with leaving the EU and blaming May's government for it.
As a result, 8 Labour MPs left and joined up as Independent MPs and Labour got tanked in the polls: https://www.politico.eu/article/poll-the-independent-group-tempts-labour-voters/
So a couple of weeks ago, Labour said they'd back a second referendum:
https://www.politico.eu/article/jeremy-corbyn-brexit-labour-backs-second-brexit-vote-to-stop-theresa-may-deal/
But now they've said today is 'not the right time'. I still maintain the cleanest way to do it - for the long term sanity of everyone involved - is to allow Brexit to happen in a controlled manner and then see if there's an appetite to change. It sounds like a bit of a piss take if you're sitting in the EU and looking in but a referendum now is kicking the can down the road till the next time the Eurozone is in crisis. Plus the good thing about May's deal is that it would make rejoining the EU easy as well.
