This trope is probably due to the fact that the majority of well-known Hollywood movies set in the UK (Mary Poppins, My Fair Lady, etc.) are set in the Victorian and Edwardian eras, when stoicism and reserve were enforced as honourable virtues by the posh boarding schools which had just become popular among the country's elite. Society had come to perceive the upcoming generations as being too mollycoddled as a result of the Edwardian and Victorian love of children and domesticity, and a culture of doting fatherhood. Thus, boarding schools were made as disciplinarian and emotionless as possible to desensitize boys and test them whether they are worthy of inheriting and perpetuating The Empire. This time was also the height of the Anglo-American "social purity" movement that denounced drinking, smoking, gambling, masturbation and prostitution as ruinous to society and organized more sedate leisure activities to help keep good people from pursuing those vices. Another, though, has to be the result of the cultural divide that has occurred since The American Revolution.
Americans, as a part of an extroverted culture of freedom of expression, are actively encouraged to express to the point that the lack of active emotional content in one's speech and mannerisms is considered quite bizarre or even psychologically dysfunctional. Thus, the more reserved and introverted culture of the British can make its people come off as acting 'fearful' or just plain robotic to Americans, who are used to their own more hammy culture. Likewise, to Brits —especially older Brits— the typical American emotional response is needlessly exaggerated at best and offensively shameless at worst.
Victorian emotional and sexual restrictions are thought to be partly reactions to the supposed debauchery of the 18th century. Where at least the upper classes were flaunting their sexual lives. Most monarchs of The House of Hanover are thought to be examples. George I (reigned 1714-1727) was thought to live openly with two mistresses: Ehrengard Melusine von der Schulenburg, Duchess of Kendal and Duchess of Munster, and Sophia Charlotte von Kielmansegg, Countess of Darlington and Countess of Leinster. George II (reigned 1727-1760) was rumoured to have several sexual escapades but only two long-term mistresses: Henrietta Howard, Countess of Suffolk and Amalie von Wallmoden, Countess of Yarmouth. George IV (reigned 1820 - 1830) had at least six notable mistresses. William IV (reigned 1830-1837) spend three decades with his mistress, actress Dorothea Jordan and only got married following her death. The exceptions were George III (reigned 1760-1820) and Queen Victoria (reigned 1837-1901) who were morally conservative, devoted to their spouses and tried to keep sexual scandals as far away from their courts and public view as possible. During Victoria's reign, the upper-classes at least attempted to keep their affairs secret in fear of scandal.
In fact, before all this came about i.e. in the 16th and 17th centuries, English people were known for being unreserved and emotional. Indeed, modern British culture seems to be heading back that way, leaving some observers to wonder if the 19th and early 20th centuries were an aberration in this regard.
