What happened in Britain, starting somewhere around 1650, was an unprecedented growth in incomes. GDP per capita in Britain went from an estimated US$1600  per capita in 1700 to $2,200 in 1800 to $5600 in 1900 (in 2011 US$), while its population grew from an estimated 8.5m in 1700 to 41.2m in 1900. There was nothing like this in the world before. Italy's GDP per capita at 1 AD, so during the Roman Empire, is estimated to have been about US$1,500, China's in 1661 at about US$940. (Figures from Maddison's long-run economic statistics, note, the normal comparative measure of economies is GDP per capita, which is a flow, not a stock, and also has a number of limitations as a measure of income, but those limitations are chiefly built around making it easy to calculate for a country with limited statistical capacity, which covers most of history and a lot of the world today - a full collection of data for national accounts is expensive). 

Before the British Industrial Revolution, the countries described as 'rich' typically had much of their population living in poverty. Famines for example were recorded in Ancient Rome and in China. Incidentally, the last peacetime famine in the Netherlands was recorded in the 1590s, before the last peacetime famine in England in the 1620s (the last peacetime famine in lowland Scotland was in the 1690s, I have not been able to find a date for Wales). So ancient civilisations just weren't that wealthy by our standards. 

As for why the struggle to catch up, well, no one in Britain planned the Industrial Revolution, it just happened. So it's not like another country's policymakers could ask the British government what they did and get a useful answer. Even now, we don't know with any rigour why the Industrial Revolution happened in Britain in particular, though there's certainly lots of theories. Plenty of European countries struggled to industrialise in the 19th century, including Italy and France. Plus, when we look at what the countries that did industrialise during the 19th and 20th centuries, the properties they tend to share are complex things like relatively lack of civil wars, and relatively low levels of corruption, and relatively high levels of social trust, which are not simple policies to implement, unlike say cutting tariffs. For example, the UK government has managed centuries without a civil war in Britain, and The Republic of Ireland has been fairly internally peaceful since the 1920s, but, despite this track history, "The Troubles" in Northern Ireland ran on for decades before the Good Friday Agreement in 1998 (and exhaustion with the violence on all sides appears to have been a significant factor in driving its settlement, as much as any skill by the British or Irish government negotiators). There is also evidence that European colonisation during this era was destructive to social capital in the subject countries, making it harder for said countries to achieve that peace, social trust and relatively low corruption afterwards. 

Non-European countries were also further hampered by the physical and cultural distance to Britain: an enterpreneurial-minded 19th century German with limited political connections would have much easier time of finding a German-English translator to work with a British mechanic than his (or her) hypothetical counterpart in Shanghai. The Anglosphere offshots of course had the advantage of close cultural connections.  

The more I learn about this area, the more amazed I am at what countries like Switzerland and Japan, or Botswana, managed to pull off. 

In short, our standards of wealth changed drastically with the Industrial Revolution and catch-up is hard. Particularly post colonisation.

Sources

Daron Acemoglu, Simon Johnson, James A. Robinson, The Colonial Origins of Comparative Development: An Empirical Investigation  The American Economic Review, Vol. 91, No. 5 (Dec., 2001), pp. 1369-14, available at https://economics.mit.edu/files/4123

Max Roser (2020) - "Economic Growth". Published online at OurWorldInData.org. Retrieved from: 'https://ourworldindata.org/economic-growth' [Online Resource]

Joe Hasell and Max Roser (2020) - "Famines". Published online at OurWorldInData.org. Retrieved from: 'https://ourworldindata.org/famines' [Online Resource]

Maddison Project Database, version 2018. Bolt, Jutta, Robert Inklaar, Herman de Jong and Jan Luiten van Zanden (2018), 

“Rebasing ‘Maddison’: new income comparisons and the shape of long-run economic development”  Maddison Project Working Paper, nr. 10, available for download at www.ggdc.net/maddison .