I think ReaperReader above is definitely correct when they ask how do you define economic power. As they suggest, by the 20th century the US economy had grown larger than the British by probably all normal methods of measurement. But that arguably doesn’t necessarily translate into power. Until the second world war I still think it is fair to say that Britain exerts a high degree of influence and control over the global economy greater than its gdp etc. British control over the mechanisms of international commerce is large. Shipping for example was predominantly registered and built in Britain until the very late 30s. In 1901 50% of world motor and steam shipping is registered to British companies (not including ships owned by dominions). This reduces to 25% in 1939, but is still great compared to 4% to 13% owned by the US. British institutions still hold the monopoly on trade arguably until the first few years of the second world war. Insurance in particular of shipping allows global trade to function. Crudely, whilst the US becomes the engine of global production, Britain still has its grip on the steering wheel and pedals probably until Britain abandons attempts to maintain a “normal” economy after the fall of France.

Britain’s ability and willingness to exert control over the global economy is something that appears quite often during the First World War and during planning for the Second. Nicholas Lambert’s Planning Armageddon, although quite sensationalist and contested, describes how Britain was considering conducting a Shock and Awe economic campaign. By the withdrawal of insurance, credit, bank transfer facilities, shipping licenses, cable communications, coupled with more traditional Belligerent Rights naval seizures and blockade, Britain was considering breaking global trade to bring German economy to halt at the start of the First World War. This would be coupled with aggressive purchase of products to deprive Germany of materials and price manipulation. What stopped them, in part, was the self destructive nature of the concept and the huge negative impact it would have on neutrals, especially the United States.

Similar plans were conceived for wars against Japan in the late 20s and 30s. Close blockade of Japan was eventually considered too difficult, and the blocking of trade from China unlikely. Instead Britain considered distant blockade off the coast of Americas interdicting Pacific trade with Japan. And like the previous plans against Germany would involve liberal spending to force-purchase materials to deny Japan and to manufacture changes to the global prices to work against Japan. Again this was deemed to have negative affects on relations with the US (an understatement if ever there was one)

In the interwar period the United States, I think, was quite aware of this predicament, especially in regards neutral/belligerent rights to trade. One of the drivers for the US navy to become “second to none” was precisely this. That the US economic growth and trade could exist at the sufferance of Britain was viewed as intolerable. The naval blockade of Germany during the WW1 alarmed the US and highlighted the control Britain was able to exert over US trade if it chose to. Additionally the threat of a closed “Sterling block” market was viewed as a grave concern, especially the fear that the eclipsed European empires would somehow club together and exclude the US from perhaps half the worlds markets was a theoretical problem to be solved. Ironically it took the Great Depression to force Britain into protectionist Imperial preference. And even then it was controversial and never as "closed" or autarkic as the US feared.

Ultimately it was the second world war that “solved” all these issues for the US. How to supplant Britain as the Wheel as well as the Engine of global economy without some form of conflict or trade war. The combination of Cash & Carry followed by Lend Lease systematically broke the British economic ability to function normally. Import and export markets fled to the US even during the war, Liberty ships caused a glut in global shipping helping to ease Britain from its position as primary carrier, and the dollar became the global lubricant of trade.

So in summary, the Second World War was the point at which US global economic dominance was incontestable. Prior to that, though overtaking Britain in most raw measurements of economic strength, could not leverage that strength into extensive global political power in part because Britain still retained part control of much the mechanisms of global trade.

Sources. From a mixture of Nicholas Lambert for the WW1 economic warfare stuff, Adam Tooze: The Deluge for US intentions post WW1. George Peden, Patrick O’Brien, Corelli Barnett (thoroughly biased, but his seething anger is an interesting view) for British global economic power, lend lease and Cash &Carry. Brian Mckercher has written at length regarding British-American interwar relations and is a great place to start for the balance of power between them. John Ferris has written a lot about British power in the 20s. And Michael Simpson on Anglo-American Naval relations in the interwar period and Things like Belligerent rights and the diplomatic issues they causes is also covered well by Stephen Roskill. Christopher Bell has written extensively about British war plans with Japan and has covered economic aspects as well.

You are right that it was a gradual process without a real date. And, by at least one measure, that of global financial trading centres, London surpassed New York by the end of the 20th century. 

The most common way of measuring economic output is GDP - a measure of what is produced in a country within a given time period. For measuring economic power, a better measure might be national income, which measures GDP plus the income earned by residents from activities abroad, minus the income earned by non-residents engaged in domestic economic production. Unfortunately it is hard to get useful data for the national income figures without the modern statistical system so GDP is the common measure - including today because a number of poor, small, countries today still struggle to even produce GDP. 

 

Economic historians have produced a number of GDP estimates for the UK and the USA before official GDP statistics were published, including annual estimates back through the 19th century. But the errors of these are large, due to the lack of standardised economic data collections. 

And, when it comes to comparing GDP between countries there's a whole bunch of new problems. How do you convert, when each country's GDP is measured in its own currency? You can use market exchange rates, but these reflect things like changes in demand for currencies (e.g at times of economic upset the US currency often rises as investors seek to move their money to safety). So economists seek to calculate purchasing power parities (PPP): the exchange rate that equalises consumers' ability to purchase a representative basket of goods. This adds a further level of error to measuring GDP per capita, and again a much bigger one for time periods before countries started collecting detailed price data to calculate things like Consumer Price Indices.

All this said, The Maddison Project is an ambitious effort to provide comparable historical information on economic output, and their database indicates that the USA surpassed the UK in terms of both total GDP and GDP per capita about the middle of the 19th century. 

But that's one meaning of economic power. London did remain the world's leading financial centre until about the end of WWII, when New York took its place and the US dollar replaced the UK sterling as the reserve currency (quite explicitly under the Bretton Woods system). That said, London started to rise again as an international financial centre from the 1970s, and in 2000 the IMF could say:

International Financial Centers (IFCs)—such as London, New York, and Tokyo—

....

In terms of assets, London is the largest and most established such center, followed by New York, the difference being that the proportion of international to domestic business is much greater in the former [London]. 

On the other hand, the USA does still have more voting rights in the IMF than any other country. 

So depending on how you measure economic power, you could say the middle of the 19th century, or about the end of WWII, or that, in terms of financial power, London (and thus the UK) was the bigger even at the end of the 20th century. Okay the last is being a bit cheeky but it gives an indication of the complexity and ambiguity of any measure of economic power.

Sources (not linked in text) .

Adam Church, The Rise-and-Fall of Leading International Financial Centers: Factors and Application, 7 Mich. Bus. & Entrepreneurial

L. Rev. 283 (2018).

Available at: https://repository.law.umich.edu/mbelr/vol7/iss2/5