London – British Prime Minister Boris Johnson is calling for national unity after his Conservative Party's landslide victory in the U.K. election. He's urging people to "let the healing begin" in the country that has been fiercely divided since the vote to leave the European Union three years ago.

The election result means Brexit will happen in January. It also means Johnson will face big challenges in keeping the United Kingdom united.

After thrashing his political opponent this week in the U.K. vote, Johnson took a victory lap – first to Buckingham Palace to meet the queen to get permission to form a government and "to get Brexit done," CBS News correspondent Ian Lee reports. He then went to northern England and newly begotten spoils: an industrial heartland that never elected his party before.

"Well I voted Boris because I believe that the people three and a half years ago who voted to leave, they should have respected that, and they didn't," one woman said.

The prime minister is also on a charm offensive, playing Mr. Nice Guy to try to heal a country deeply polarized by Brexit, which now seems inevitable.

"I want you to know that we in this one-nation Conservative government will never ignore your good and positive feelings of warmth and sympathy towards the other nations of Europe," Johnson said.

As the prime minister prepares to leave the E.U., trouble is brewing for him on another front. Scottish nationalists won a resounding victory and are now demanding their own independence vote from the U.K.

"This is not about asking Boris Johnson or any other Westminster politician for permission," Scottish National Party leader Nicola Sturgeon said this week. "It is instead an assertion of the Democratic right of the people of Scotland to determine our own future."

Johnson already ruled out another Scottish independence vote. But there's potential trouble for him elsewhere. Nationalists won in Northern Ireland, too, possibly paving the way for a vote on Irish reunification.