The Unite leader, Len McCluskey, has stated his willingness to be a “critical friend” to Nicola Sturgeon’s SNP government, while urging the Labour partyto apologise for the party’s “betrayal” of Scottish voters.

McCluskey told delegates at Unite Scotland’s first policy conference in Clydebank, West Dumbartonshire, on Sunday: “Nicola Sturgeon and her team have reached out to trade unions – including on vital issues like blacklisting – and we would be letting our members down if we responded anything other than enthusiastically.”

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While insisting that “Unite remains a Labour union, here in Scotland as across all of Britain”, he cautioned: “Preferring a Labour administration cannot mean being blind to reality, or ignoring the opportunities that we have to advance Unite members’ interests.”

Acknowledging that more than two-thirds of Unite Scotland members voted for the SNP in last May’s general election, McCluskey added: “Being a friend does not mean being an uncritical friend. We can and should demand more from the SNP. Nicola’s government should not be hiding behind procedural niceties in relation to the trade union bill.” (The Scottish government recently failed in its attempt to secure a legislative consent motion in Holyrood, which would have allowed MSPs to vote against the bill’s application north of the border.)

McCluskey, who will meet the first minister and SNP leader for the first time later on Sunday, added: “I’ll be saying to Nicola when I meet her later – don’t just oppose this wretched bill, but block it in Scotland. And while you’re at it – end the council tax freeze and really go the extra mile to lift the cloud of austerity from the lives of the people of Scotland.”

Speaking at the conference on Saturday, the Scottish Labour leader, Kezia Dugdale, likewise called on trade unionists to campaign with her party against the SNP’s cuts to local government.

But McCluskey suggested that Dugdale must also apologise to Scottish voters who were alienated by “the ideology of New Labour”.

Speaking to journalists before his conference speech, he explained: “The ideology of New Labour effectively alienated large swaths of the Scottish working class, which manifested itself quite dramatically last May. Kezia [Dugdale] has to effectively say: ‘Labour is under new management, we apologise for betraying you, and we will start from scratch to try and build that trust up.’’

Revealing that internal polling had found 65% of Unite Scotland members supported the SNP at the last general election, compared with 80% voting Labour historically, he said: “The SNP stole most of the radical clothes that historically should have belonged to Labour. The truth of the matter is that in Scotland the SNP seemed to lots of people as a more social democratic party than Labour.”

But the Unite leader also warned that both Dugdale and Jeremy Corbyn needed time to win back Scottish voters.

McCluskey said: “Some have started to talk about the May elections as a referendum on both Jeremy Corbyn and Kezia Dugdale. What a lot of nonsense. From all the opinion polls it looks like the SNP are going to make further gains in May, but for that to be laid at the door of Kezia Dugdale or Jeremy Corbyn is just plain nonsense.

“The task in Scotland almost makes Mount Everest look like Ben Nevis, so she’s got to be given time and support and Unite will be playing our role in that.”