The Scottish and Welsh leaders said the Prime Minister should urgently change course and push for a “Norway plus” deal to avoid economic disaster.
Ms Sturgeon and Mr Jones urged the Prime Minister to ignore Leave supporters in her party and abandon her Brexit ‘red lines’ – which include leaving the bloc’s single market and customs union.
They said Mrs May’s strategy means Britain was being railroaded into a hard Brexit, which they said “will deeply damage our economies and possibly jeopardise our security”.
And their warning has been timed to coincide with today’s meeting of the British-Irish Council (BIC) in Guernsey, where the pair will meet with Mrs May’s deputy, David Lidington, and Irish Prime Minister Leo Varadkar.
In a joint statement, Ms Sturgeon and Mr Jones said: “In practice, the Prime Minister’s position on issues such as customs arrangements and regulatory alignment appears increasingly difficult to reconcile with the red lines, but she cannot come clean because she is held hostage by the Brexiteers in her Cabinet and party. This cannot continue.
“We call on the UK Government in its forthcoming white paper to commit to staying inside the single market and customs union recognising that this will require continued alignment with the EU regulatory environment.
“The aim should be a ‘Norway plus’ model on the basis that the red lines set out by the Prime Minister in her Lancaster House speech in January 2017 are not consistent with the national interest.”
This latest move comes amid a souring of relations between Westminster and Holyrood.
Ms Sturgeon’s Scottish National Party (SNP) has been outspoken in its opposition to Brexit and the First Minister has accused the UK Government of treating Scotland “with utter contempt”.
The deep divisions were highlighted last week when SNP MPs staged a Commons walkout in protest of over the handling of the EU Withdrawal Bill.
But Mr Lidington insisted the UK Government is committed to maintaining cordial relations with Scotland and Wales.
He said: “Working together is exactly what people all across the UK want to see their respective governments doing.
“They want to see their lives being improved through joint endeavour. They don’t expect different governments – with politicians from different parties – to agree on everything.
“But they do expect us to keep talking – and listening – to each other, and to co-operate on the big issues that affect us all.
“The UK Government, for its part, is committed to keeping up the pace, and improving the depth, of our co-operation.
“Whatever the inevitable political noise as we unpick the complexities of our membership of the EU, I will ensure that we maintain the depth of our ongoing engagement with the Scottish Government, including this week at the British-Irish Council in Guernsey.
“I hope that the Scottish Government will maintain their own side of that relationship.”
A UK Government spokesman said: “When we leave the EU the whole of the UK – including Wales and Scotland – will be leaving the customs union and single market. There is no change to that position.
“Leaving the customs union means for the first time in 40 years, the UK will have the freedom to strike our own trade deals.
“Our focus is on getting the best deal possible for the UK – one that allows us to take back control of our borders, laws and money.”