More or less rigorous attempts to describe or calculate the size of national economies go back a long way, to the European enlightenment and before. Most of these seventeenth and eighteenth century efforts were made on behalf of heads of state or government to support the expansion of state power or underpin war efforts through identifying potential for taxation or improving the efficiency of economic sectors. Yet such disparate efforts at national accounting were hampered both by the lack of reliable macroeconomic data and by opaque methodology and definitions. It is difficult to calculate the size of a domestic economy if accounts are lacking or inconsistent, and it is impossible to use the result for any meaningful comparison if it is unclear how the result was arrived at in the first place. Progress towards solving these problems was made from the late eighteenth century onwards, beginning with the proliferation of and debate on Adam Smith’s ideas on methods for calculating national wealth and income (although these still differed notably from how such calculations are made today), and the increasing sophistication and comprehensiveness of national statistics championed by economists such as Alfred Marshall.
The modern definition of Gross Domestic Product has its origins in the efforts of Colin Clark. Clark was a British economist who through the 1920s and 30s sought to use statistical data to calculate comprehensive quarterly estimates of national income and expenditure. Clark was not the first economist to attempt this, but his work set a new standard for quality. His work also came at an opportune time. The Great War and the economic crises of the interwar era legitimised a greater degree of government intervention in and regulation of economies, and the demand for economic data which could be used to direct such intervention was growing rapidly. Beginning in 1930, the British government formally adopted Clark’s estimates as a basis upon which to formulate economic policy. Clark’s methods were adopted and refined further by the Russian-American economist Simon Kuznets, and from 1934 the resulting estimates were used by the US government to formulate economic policies under the umbrella of the New Deal programmes.
Further elaboration of the accounts pioneered by Clark and Kuznets would come during WWII. The expansion of government expenditure brought about by the war prompted the inclusion of government expenditure in the calculation of domestic production. This was a major revolution in the evolution of national accounting. In the words of Diane Coyle
For two centuries, “the economy” was the private sector. Government played a small role in economic life, and featured mainly because it looked to raise taxes to pay for wars.
The massive expansion of state economic activity, beginning in the nineteenth century and growing exponentially from 1914 onwards, had made this assumption untenable. Government spending could and was having a massive impact on the overall economy of nations. The point that government income and expenditure must be seen as an integral part of the domestic economy as a whole was touched upon by John Maynard Keynes in his 1939 work The General Theory of Employment, Interest and Money, and then made explicit in his 1940 exposé How to Pay for the War. Similar developments were taking place concurrently across much of the industrialised world, including in the Netherlands, Germany and the Soviet Union. As a result of his proposal, Keynes was put in charge of a new British statistical agency and given the task of producing economic data from which wartime government policy could be derived. Two other economists at the British Treasury, Richard Stone and James Meade (both would later be awarded the Nobel Prize in Economics), were spearheading similar efforts to comprehensively incorporate government spending in national accounts. Together Keynes, Stone and Meade would produce what can be considered a form of proto-GDP estimates, published from 1941 onwards. The US Department of Commerce would create and publish similarly expanded estimates (then known as GNP – Gross National Product) beginning in 1942.
Following the end of the war efforts were made to harmonise the advances in accounting made during the interwar era and through the war, so as to facilitate international cooperation on economic issues through Bretton Woods and the United Nations. These efforts, soon concentrated under the aegis of the newly created United Nations and led – at least initially – by British and American economists, culminated in the publication of a unified United Nations System of National Accounts (SNA or UNSNA) in 1953. The SNA established standards for the creation of estimates of aggregate domestic economic activity, creating the first international definition of Gross Domestic Product. Although the SNA has been continuously revised and updated since, it remains the international standard for calculating GDP.
It's not really a big thing, but I would nitpick your wording a little. GDP rather represents income (or as the name actually says - production) than wealth. A nation could've accumulated a lot of wealth, but fall behind in GDP to a nation with less accumulated wealth, but more current production/income/GDP. GDP and wealth are obviously highly correlated, but important differences come out in some situations. For example let's imagine a hurricane/natural disaster wrecking a nation. It will destroy a lot of wealth, but might cause an increase in GDP, as new loans are taken, new constructions begin and expansionary economic policies are used.