The realization that Britain being alone after the fall of France is a myth is not a new phenomenon. Much of what I am writing now borrows from the 2018 A.B. Emden Lecture at St. Edmund’s Hall, Oxford, by Professor David Edgerton who talked about this very idea. Though with all enduring ideas, the popular perception of fortress Britain in WWII is in part true and its more mythical elements comes from a distortion of history due to the passing of time. To put it briefly, there are two main elements to the question you are posing. First, is the idea of Britain being alone historically accurate given their massive empire? And secondly, if that is not the case, how did we come to see an isolated Britain in the early parts of WWII as the prevailing popular history narrative?
To address the first question of how overmatched Britain was after the fall of France, perhaps the first thing we should address is that while the Empire was a source of strength for Britain, it was also a massive liability. While Britain could draw upon manpower and materiel from its globe-spanning empire, it also meant that it had defence obligations that spanned a global empire. In this sense, the Empire was very much a double-edged sword since channelling forces from overseas into specific theatres would leave them at best underdefended. The root of this problem can trace itself back to the First World War. Even though Britain was fighting alongside France in a very similar strategic position, it still cost them an inordinate amount of money and lives to prosecute the war. While this did not cripple Britain economically, it was such a great strain that it evaporated any future political will to maintain that level of spending unless it was absolutely necessary. This was reflected in their inter-war defence arrangements which saw them become begrudging signatories to the Washington Naval Conference and following that other disarmament and arms limitation treaties. The curbing of military spending in the interwar period drastically affected the disposition of British and Commonwealth forces as spending was largely limited to as the least possible amount which was deemed to be acceptable. This was manifest in the “10 Year Rule” where the government maintained after WWI that it would not increase defence spending for the next decade.
The Royal Navy perhaps suffered the most under these budgetary cutbacks since their responsibilities were the largest. The sheer size of the Empire meant that the Royal Navy was essential in defending far-flung territories and maintaining lines of communication with them. This is an essential component in order to realize any advantage of an Empire. In the event of a major conflict, troops would have to be redeployed or mobilized from the Dominions and resources raw or manufactured would have to all be transported via shipping. In addition to this, the Royal Navy would have to direct combat enemy fleets and undertake their own offensive action. The Royal Navy was after the war reduced to a “One Power Standard” which meant that it was only maintained to be as large as their nearest competitor which was the United States. This meant that the Royal Navy was stretched incredibly thin, even considering that a conflict with the United States was relatively unlikely. The strategic position also changed by the early 1930s with Japan becoming a significant threat to the Asia-Pacific region of the British Empire. This combined with the naval build-up in Germany and Italy meant that the Royal Navy was forced to draw up their initial war plans relying on the French fleet to contest the Mediterranean so that they could maintain adequate forces in the Far East and Atlantic.
This whole plan fell apart with the swift defeat of France in 1940 which created at the very least an uncomfortable strategic position for Britain. While scholars like Richard Overy had made a very convincing case that Britain was not in danger of being invaded, they were very much on the backfoot even though they were not at the brink of the defeat. Britain by this point was also mobilizing its entire Empire for War. Commonwealth and Exiled forces from continental Allied nations were taking part in the Battle of Britain and in North Africa, ANZAC and Indian troops had been deployed. The United States while being official neutral was by this point clearly sympathetic to the Allied cause, supplying them with Lend-Lease material.
The difficulty in the British position came from the dispersion of the resources they were drawing on which were relatively isolated from the British Isles. This meant that controlling in the Atlantic, North Africa and the Mediterranean was essential to maintain supply lines but were themselves difficult to supply from Britain where a majority of war materiel was being manufactured. All this is to say that the British position was precarious, but they were not at risk of being defeated outright. This is, of course, said with the benefit of hindsight. While it was apparent by the end of 1940 that the Germans were unable to mount a successful invasion of Britain, there were still fears of it and it tied down a considerable amount of resources. In addition, while the U-Boat threat and even potentially the loss of North Africa would not necessarily be death blows to Britain, there was considerable worry that these losses would compel them to surrender. More as a matter of losing the political will to carry on the war than the sheer untenability of their position.
What did spell disaster for the British was the Japanese entry into the war which captured the entirety of Southeast Asia as well as threatening an invasion of India and Australia. Again, with hindsight, it is clear that the Japanese had overextended themselves and their invasion of Burma was the highwater mark of their military capability. However, at the time the situation appeared to be incredibly dire having swiftly lost the Malayan Peninsular in a shocking campaign, Darwin suffering an air raid in what seemed to be a prelude to invasion, as well as the British fleet in the Indian Ocean suffering significant damage due to a Japanese raid. Again, the situation is similar to that of earlier in 1941 in that the British position was precarious but they were not close to being destroyed. They had suffered significant losses but the sheer size of the Empire itself also meant it was difficult to physically deal a crippling blow. Fortunately for the British, the Japanese entry into the war also drew the United States into the conflict which soundly tipped the long-term balance of power in favour of the Allies.