I just can't figure out why the British Empire's economy was in such trouble despite having so many colonies and possessions.
There are a variety of reasons why Britain struggled in the 1920s, but the clearest explanation is monetary. British policymakers, most famously Winston Churchill, decided to return to the pre-war gold parity for the pound sterling, at a time when other countries had instead permanently devalued. From the standpoint of late 20th-early 21st century central banking orthodoxy, this decision was not just poor, it was downright perverse. But, as I will explain below, it reflected 19th century rather than 21st century logic on monetary policy, for obvious reasons, and if some (Keynes being the most famous) warned about its consequences, his was the dissident and not the orthodox position. The combination of postwar chaos, the Spanish Flu and monetary policy mistakes created a severe (and international) depression in 1920-21, but it was needlessly tight monetary policy that meant the UK did not entirely recover until after WWII.
Britain had been the centre of the pre-war international gold standard. The Bank of England's role in managing that system has been somewhat exaggerated, but it was nevertheless the most important hub of international finance. The war had forced Britain to break the gold parity, and there had been considerable wartime and post-war inflation created both by scarcity of consumer goods, and pent-up demand. Adhering to the gold standard during wartime was simply too costly, and the war was too much a priority to sacrifice at the altar of hard money. - only the US among major powers stayed on the gold standard in wartime. And so, the value of the pound drifted downwards in terms of domestic goods (that is to say, there was inflation).
Returning to the pre-war gold parity was seen as critical for restoring investor confidence in the pound, in UK sovereign debt, and in the UK's place in the financial world more generally. To accept the devaluation of wartime as a permanent rather than temporary measure would be both humiliating, and contrary to the central banking orthodoxy of the time: that confidence in gold convertibility was the mark of a well-managed economy. Britain was under domestic pressure to reject the gold standard orthodoxy, but this pressure was not as strong as elsewhere, and the establishment not as weak. In the UK, the Labour Party had brought the voice and issues of the working classes into Parliament, but the centre(-right) nevertheless held, and the government did what was necessary to return to the pre-war parity. Britain endured deflationary pressures, and successfully (though only briefly) re-established their domestic gold standard at the pre-war level. The enormous cost, in terms of unemployment and reduced export competitiveness, was borne by workers and taxpayers.
However, several crucial things had changed, making the interwar period a fundamentally different (and more unstable) environment for monetary policy. For one, the relative strength of the British economy had changed. Prior to WWI, one could perhaps argue that the British economy was at least approximately the equal of the American economy, and had its own unmatched successes: it was still obviously the centre of international finance, and its investors owned an enormous portfolio of stocks and bonds representing much of the world's business and infrastructure, not to mention the debts of most sovereigns. The UK balance of trade was generally in deficit, but its balance of payments was in surplus: that is to say, the UK relied on investment income to support the value of the £. The war put enormous pressure on UK finances, and much of this portfolio was liquidated in one way or another. The Americans, meanwhile, did not suffer this setback, and increased their share of international finance continuously.
British exports suffered from increasing competition during this period. From the 1840s until the late 19th century, British manufactured goods and services exports were a a very large component of world trade, and successfully competed even with domestic production across the world. By the end of the 19th century, Britain's advantage was eroding for a number of reasons: domestic production was slowly industrialising all over the world, including in the British colonies; Japan was beginning to compete in Asian textile markets; other European countries had caught up in terms of industrialisation, and had even outstripped British productivity levels in some industries (Germany in most heavy industries, for instance), though they maintained a lead in some of the lighter industries. The international trade and finance that Britain had intermediated recovered only slowly, and other countries competed both economically and politically in a way that had barely begun prior to WWI.
In terms of domestic politics, the countries of Europe had been transformed. While the beginnings of mass democracy, the labour movement, the welfare state, women's suffrage, and so on predate WWI, there is no question that the war permanently transformed both who had voice in politics, and what could be said in the political arena. The old liberal oligarchies found themselves challenged by new parties with different priorities. Whereas previously any issue that could unite both the liberal and conservative factions of the upper and middle classes would be taken for granted, such as balanced budgets and adherence to the gold standard, now these issues were all contestable and contested. Labour movements and unionization had fundamentally changed the nature of the wage bargain, and increasingly deployed their political, social and economic power towards the questions of employment, working hours, labour conditions, wages and benefits. Socialist parties across Europe were organising, participating and in some cases actively rebelling against the existing political order. The example of Russia validated both hopes for radical change and fears of it. And so, what might have been a painful but politically obvious decision to restore the pre-war economic status quo was instead a turbulent and messy bargain within and between countries, which would not last.
The extreme example is the chaos of Weimar Germany, where the failed Spartacus League uprising was just the beginning of two of the most turbulent and tragic decades in European history. Reparations, war debts both internal and external, and rising political violence meant Germany was understandably unwilling (perhaps even entirely unable) to return to the old orthodoxy, with stable currency convertible to gold and regular debt repayments. Any government that satisfied international creditors and made a serious attempt to balance the budget, as in the case of Bruning's chancellorship, would render itself politically toxic. "Internal devaluation" meant lower wages, rollbacks of benefits, increased unemployment, and general misery. This did not last, and the consequences of the attempt almost certainly include the rise of the Nazi party to power. France was not quite so unstable, but nevertheless did not return to the pre-war parity, and instead devalued the Franc by 80% (!).
All of this meant that Britain's attempt to return to the pre-war gold parity was far more than just a restoration of the status quo. The relationship of the general price level to the price of gold is only stable if both the scarcity of gold and the scarcity of foreign currency remain stable compared to the domestic money supply. Neither of those things held true. Gold production could not keep up with the enormous pace of global economic growth, especially in the United States. This caused gold prices to increase (which is deflation, if the gold-money parity is maintained). French devaluation led to the Banque de France accumulating an enormous store of gold, well beyond the level justified by the size of their economy. Thus, what seemed like the merely a conservative and common-sense restoration of an old and stable rule ("the £ can be converted into a fixed amount of gold") was instead a Quixotic overvaluation of the currency compared both with the domestic price level, and the new, much lower prices of foreign currencies.
The relationship between colonies and economic development is less clear, but in the short run, what could have been done to somehow leverage the Empire to solve any of this is not clear. The richer dominions had their own economic problems to deal with, and Britain had long since given away the power to levy whatever taxes they pleased on Canada or Australia. The poorer colonies in India or Africa did not have much surplus to expropriate, as they were already quite close to the level of subsistence. Extracting large amounts of revenue from very poor people is difficult, expensive, and largely self-defeating. Even the most rapacious colonial policy would almost certainly not have yielded economically relevant amounts of plunder, compared with the sheer size of Britain's industrial economy by the 1920s.