Why was Britain the first country to industrialise in the 18th and 19th centuries?

Good question! There is still a lot of uncertainty about why industrialization happens when and where it does (and doesn't), but there are a few unique features of 18th and 19th-century Britain that historians and economists have found to be important in triggering industrial development and economic growth.

The first and probably most important is Britain's relatively pluralistic, inclusive institutions. While they were a monarchy (I mean, technically they still are), Britain had a strong tradition of democratic representation, public dialogue, and individual rights. The monarchy's power was not absolute, but limited by Parliament and judicial precedent enshrined in common law. Individual people in Britain were relatively free to own their own property, make their own decisions, and generally live their lives without excessive government interference. This allowed for more free markets and more productive workers, as those who were productive could keep and sell what they produced instead of having it confiscated by the government, and therefore had more incentive to produce it in the first place. This is not to say that Britain didn't have taxes and regulations, which of course they did; nor that all taxes and regulations are bad---many are necessary, and others can be beneficial. But all other things equal, lower taxes and more permissive regulations are generally associated with stronger economic growth.

Another important reason was Britain's natural resources, specifically its large forests and extensive coal reserves. Industrial technology is really only useful if you have the fuel to power it; at that time petroleum was still very difficult to refine, and obviously nuclear fission and photovoltaics wouldn't be invented for over a century; so burning wood and then coal was the main way that large machines could be powered. Britain had these resources in abundance, which made it cheaper for them to industrialize than it was in most other places.

Britain also had a relatively high literacy rate, meaning they had more skilled artisans, inventors, and engineers per capita than most other places. They had a strong national defense (the British Navy was the mightiest in the world) and thus were safe from outside invasion. They had a relatively low crime rate. Their currency, the pound, was very strong and very stable (indeed, still is and always has been). All of these factors likely contributed to Britain's industrialization before other places. The United States was not far behind, perhaps because it shared most of these features with Britain as well as sharing many features of British culture and institutions.