Scotland’s first minister, Nicola Sturgeon, has written to Boris Johnson to tell him that she is “looking forward” to discussing with him her proposals for a second independence referendum.

Using her first letter to the new prime minister to remind him of Scottish government analysis which found that a no-deal Brexit could cost 100,000 jobs across Scotland, she writes: “Given your public comments about leaving the EU on October 31 with or without a deal, “come what may” and “do or die”, it is now – more than ever – essential that in Scotland we have an alternative option.”

Stating that her government would continue the parliamentary progress of legislation to enable a second referendum on independence, which Sturgeon has indicated she would like to hold before 2021, she tells Johnson: “The right of the people of Scotland to determine their own future is a basic democratic principle that must be respected.”

The referendums (Scotland) bill does not specify the date, question or referendum period, which would all be set by secondary legislation. The Scottish National party leader has previously committed to securing the necessary transfer of powers from Westminster.

Johnson refused to rule out blocking a second referendum on Scottish independence, even if the SNP won a mandate for one at the next Holyrood elections, when he visited Scotland during the Conservative leadership campaign, maintaining that “we should stick to that promise” that the 2014 vote was decisive for a generation.

Giving Johnson a timetable for future discussions, Sturgeon writes: “Similarly any decision of the Scottish parliament on whether to give people that choice must be respected. The parliament will consider the necessary framework legislation for a referendum after the summer recess, and I look forward to taking this matter forward with you once MSPs have had the opportunity to debate the issue further.”

The letter comes after the SNP shadow leader of the house, Pete Wishart, described Johnson’s new cabinet as the “worst since Thatcher” on Wednesday night, while his fellow SNP MP Tommy Sheppard dismissed the Scottish Conservative leader, Ruth Davidson, as a “busted flush”, after Johnson apparently ignored her advice to keep the former Scottish secretary David Mundell in post.

Sheppard said: “Ruth Davidson is a busted flush – Boris Johnson has humiliated her and left her authority non-existent following the sacking of David Mundell as Scottish Secretary.”

Davidson, who has made no secret of her reservations about Johnson and said on Tuesday that she would judge him by his actions once in Downing Street, issued a warmly worded statement after Mundell’s sacking, while many other senior Scottish Tories indicated on social media that they felt his departure was a significant loss to the party.

While critics have warned that a Johnson premiership could boost support for the SNP and independence, Scottish Conservative supporters of the new prime minister have insisted in recent days that he is willing to take guidance from colleagues based north of the border.