As things currently stand, Andy Burnham will just flat-out reject their demands. The only way I can see him agreeing to another referendum is if the next general election leads to a hung parliament with Labour as the biggest party.

A better question would be if such a referendum would succeed.

Scotland: probably not. But then again, one of the biggest reasons the Scottish people voted to remain twelve years ago was due to them losing their EU membership by going independent. They could apply to join the EU, but it's very possible that Spain could veto their accession on the basis that they don't want to give legitimacy to the Catalan independence movement. Brexit is a game-changer that could make Indyref2 succeed. And Hungary under Viktor Orban could have rejected it because he was a useful idiot to Russia. What doesn't help Scotland's case for independence is that the SNP nosedived in popularity since Nicola Sturgeon resigned.

Wales: definitely not. Support for Welsh independence is actually pretty damn low. Plaid Cymru winning big in the Senedd moreso represents how badly Labour fucked up under Keir Starmer. Welsh voters tactically voted them in to prevent a Reform UK majority.

Northern Ireland: maybe. The gap has narrowed since Brexit, and there is a particular clause in the Good Friday Agreement where the Secretary of State must agree to hold a referendum to unite Ireland if it appears likely that such a vote would succeed. I think the reason why this hasn't happened is because our deal with the EU has largely shielded Northern Ireland from the negative consequences of Brexit via a backstop.