Never in his life - before the traumatic events of 1936 - did Bertie believe he would be king. He was the Duke of York, the second son of George V. His brother David, the Prince of Wales, would take the throne when their father died.
But ... there were signs of trouble even before David fell in love with a twice-divorced American (named Wallis Simpson) whom he wished to make Queen. There were indications David would want his own way, even if it meant going against the will of the people and Parliament.
Christopher Hibbert summarizes David’s personality, and its impact on people:
Restless, impatient, impulsive, frustrated, emotional, affectionate, unstable and indiscreet, he earned as Prince of Wales much popularity and some respect. He had far more charm than talent, far more capacity for feeling than for thought. His sudden enthusiasms, his discordant ambitions, were likely to die as quickly as they had been born ... It was his tragedy that he never entirely succeeded in reconciling, as his grandfather [Edward VII] had been ultimately able to do, his love of life with his position and responsibilities. (Christopher Hibbert, The Court at Windsor, page 275.)
The King himself hoped that David would never have children, so the monarchy would pass to Bertie and his heirs:
...King George’s private opinion - expressed on more than one occasion - [was] that his eldest son would not fulfil his destiny, and that the prospect of his second son becoming King had given him pleasure. (A Spirit Undaunted, page 103.)
George V would have found no pleasure in the way David handled things after his father died on January 20, 1936. Within days after David became King Edward VIII, his own private secretary contemplated a “disaster,” writing in his diary:
...increased responsibility may work a miracle, but I don’t think he will last very long. One could prop up the facade for a Prince of Wales - not so easy for a King. (Robert Rhodes James, quoting Godfrey Thomas, in A Spirit Undaunted, page 104.)
Historians believe the Prince of Wales had hoped to marry Mrs. Simpson before his father died, thereby eliminating the problem of whether he should (or shouldn’t) marry her after he became king. Still single when he took the title Edward VIII, however, his future marriage (and his resulting status) became an issue for the government (especially when his relationship with Mrs. Simpson was first reported in British newspapers).
Stanley Baldwin, then Prime Minister, flatly opposed the King’s plans. So did the Cabinet and other Dominion governments (although Winston Churchill, improbably, was on the King's side). Baldwin told Edward VIII he had three options: end his relationship with Simpson (the link is a BBC video biography); marry her against his ministers’ advice (they would then resign); or abdicate (meaning, the King would no longer be King).
Pondering what to do, David ignored the man who would become king if Edward VIII left the throne. Refusing his brother’s calls, David (who had been close to Bertie and Elizabeth before Wallis Simpson entered his life) had to decide what he wanted more - to be king or to be Wallis’ third husband.
Bertie and Elizabeth worried, greatly, about the impact of an abdication. If Edward VIII resigned, the Duke of York would have to take his place. He was neither trained for, nor interested in, the job.
On the 9th of December, 1936, David made a decision. The following day, he would sign an Instrument of Abdication which his close friend, Sir Walter Monckton, would prepare. On the 11th of December, he would tell the people via radio broadcast.
Bertie and Elizabeth were now King George VI and Queen Consort. Their official coronation would take place on May 12, 1937. That meant another radio broadcast - heard around the world - this time by the new King. Lionel Logue, who for years had checked the Duke’s speeches - exchanging hard-to-say words with easier-to-say replacements - was present for the broadcast.
By the time it was finished, people everywhere wondered why anyone had ever reported the new King had a stammer.