Humza Yousaf was sworn in as Scotland's first minister on Wednesday, becoming the first Muslim leader of a government in western Europe, but is already facing unrest in his party.

At 37, Yousaf is also the youngest leader yet of the Scottish National Party (SNP), and is vowing to reinvigorate its flagging campaign for independence.

But after he won the race to succeed the long-serving Nicola Sturgeon on Monday, his leadership was already being questioned after defeated rival Kate Forbes refused to serve in his cabinet.

The outgoing finance minister was offered a more junior role by Yousaf, despite coming close to victory. She ended with 48 percent of the preferential votes of SNP members to his 52 percent.

Yousaf's allies claimed it was because Forbes wanted to devote more time to family life after recently giving birth. But in press reports, her supporters were scathing about the job offer.

Yousaf was spending the rest of Wednesday rounding out his cabinet after he was sworn in by Colin Sutherland, the lord president of Scotland's highest court, the Court of Session.

The new first minister vowed to “well and truly serve His Majesty King Charles” III, despite his stated support for abolishing the monarchy in favour of an elected head of state for Scotland.

UK Prime Minister Rishi Sunak congratulated Yousaf in a phone call shortly after the new SNP leader was confirmed as first minister by the Scottish Parliament on Tuesday.

Yousaf said the call had been "constructive" – but that he had stressed to Sunak that "the democratic wishes of Scotland's people and parliament" should be respected by London.

Sunak stressed instead that the two governments should work together on day-to-day policy matters, according to Downing Street. (AFP)