With the Molotov-Ribbentrop pact holding and America not in the war, the British Commonwealth was virtually alone in October 1940, still fighting the Battle of Britain (albeit the final throes) when Italy invaded Greece. Although the immediate threat of a German invasion of Britain passed, the Luftwaffe continued heavy attacks into 1941 holding down the RAF (the aircraft eventually despatched to Greece were aged Gladiators and Blenheims, Spitfires held back for the defence of Britain), and the army were engaged in North Africa as well as re-arming after losing a massive amount of heavy equipment in France ("We had lost the whole equipment of the Army to which all the first-fruits of our factories had hitherto been given. Many months must elapse, even if the existing programmes were fulfilled without interruption by the enemy, before this loss could be repaired." Winston Churchill, The Second World War). There just weren't the resources to support Greece at that point, especially as it wasn't a strategic priority ("Loss of Greece and Balkans by no means a major catastrophe for us, provided Turkey remains honest neutral." ibid)

In early 1941, with the Italians heavily defeated in Libya and stalled in Greece, the situation seemed to have slightly improved. With Germany preparing to move into the Balkans, and the threat of Rommel and the Afrika Korps not fully appreciated, the British despatched what they could from Egypt ('W' Force), weakening their position in North Africa in the process. The hope was to galvanise a combined Balkan effort: "It was not what we could send ourselves that could decide the Balkan issue. Our limited hope was to stir and organise united action. If at the wave of our wand Yugoslavia, Greece and Turkey would all act together, it seemed to us that Hitler might either let the Balkans off for the time being or become so heavily engages with our combined forces as to create a major front in that theatre." (ibid) With Turkey remaining neutral and the rapid fall of Yugoslavia, that faint hope was crushed.

Initially - when the Italians invaded - the Greeks didn't want British support, they feared retaliation by the Germans in response. After a change of leadership, and the subsequent German invasion, they finally allowed the British in, but even then, it was not enough to combat the superior German forces.

A quick note before I go into detail. It was hardly a "British" defence. The majority of Allied troops in Greece in 1941 were Greek (the Greek army had about 400,000 men at the time). And a sizable portion of the other troops involved in Greece's defense were from the Commonwealth (the 6th Australian Division and the New Zealand 2nd Division both played major roles in the defense of Greece).

The fact Greece fell in less than a month can be chalked up to a variety of factors.

First off, the Greek Army was woefully unprepared for war. Greece wasn't an industrial nation, and that meant it didn't have adequate artillery, transport, small arms, or ammunition. Greek troops had been fighting since August 1940 (when Italy had invaded northern Greece) and the fighting had further depleted Greece's already limited supplies of war material. So when the Germans made a concerted, mechanized attack on Greece, the Greeks simply didn't have the resources to repel them.

Secondly, the Axis powers were able to establish air superiority over Greece. Italy's Regia Aeronautica and the German Luftwaffe effectively ruled the skies over Greece in 1941. Stuka dive bombers interdicted Allied supply convoys, harassed retreating troops, sunk ships, hammered strongpoints, and generally made life miserable for the Allies in Greece. Germany Bf109s and Bf110s quickly swept the handful of RAF Hurricanes and Gladiators from the skies. The loss of air superiority left Greek and British Imperial troops at a severe disadvantage in the defense of Greece.

Thirdly, the German and Italian forces invading Greece badly outnumbered the Greek and British Imperial defenders. This allowed them to outflank, encircle, and overwhelm Allied troops and allowed them to keep up a constant pressure on the Allied defenders.

Fourthly, there were major strategic failures on the Allies' part. The Greeks had serious gaps in their defensive lines (especially in the north-west) and the Germans were able to exploit them during their invasion.