And no, this isn't about crazy Republicans.
Towards the end of Operation Compass, the Italian 10th Army was withdrawing from Benghazi due west along the coastal road. The British decided to cut them off, and sent an armored force through the desert, whose advanced elements reached the coastal road at El Agheila barely 30 minutes before the leading Italian formation. Despite an attempted counterattack involving all of 10th Army's available armor two days later, the British held their ground and thus forced the surrender of all but a few thousand of 10th Army's men - a total of 25,000 prisoners, 107 tanks and 93 guns.
As this was taking place, Churchill had decided to send a large part of his available forces to Greece. Eventually, the 6th and 7th Australian divisions, the Polish Brigade and one armored brigade from the 2nd Armoured Division were sent.
At the same time, as part of Op. Sonnenblume, the DAK had begun arriving in Africa, although under orders not to engage in attacks untill the transfer of troops was complete. Thanks to ULTRA intercepts, the British were aware of this, which is why they didn't expect any attack any time soon.
Rommel however correctly recognized that the deployment to Egypt had left the already strechted Empire forces in a very weak position. Cyrenaica at the time was defended by the 9th Australian division, with two brigades near the coast south of Benghazi, and by the exhausted 2nd Armoured Division, who had taken to requisitioning Italian tanks and guns to make up for the shortfall in equipment. Moreover, the latter formation was tasked with defending the 9th Division's flank, which forced it to operate from a series of depots strung out across the area, as there weren't enough trucks and Benghazi harbor wasn't yet operational. Lastly, the 3rd Indian Motor Brigade was available as a sort of reserve, based west of Tobruk at Martuba, altough it had just arrived in Cyrenaica and had no tanks, artillery or anti-tank guns and only half its wireless sets.
OTL, Rommel attacked on the 24th and caught the British completely off guard, forcing a general retreat. Like the British almost two months before, he had sent part of his force along the coastal road (which ended up capturing Benghazi, filled with suplies and Italian POW's), whilst another part advanced through the desert in an attempt to cut off the British. While some results were achieved (a few thousand British POW's; British commander O'Connor captured; 2nd Armoured beaten up), the main prize eluded him, as the 20th and 26th Brigades from the 9th Australian Division managed to escape with the bulk of their forces to Tobruk. There, they joined up with the 24th Brigade (which had been garrisoning Tobruk) and with the 18th Brigade (which had been pretty much the only formation available which the British could transfer west from Egypt, despite it still being in the process of training), and together, they defended Tobruk from German attacks until eventually relieved months later.
This retreat from Benghazi towards Tobruk by the Australians came to be known as the "Benghazi Headache".
However, let's imagine the following situation:
The British retreat is slowed down on account of weather, more inept organization, enemy air attack or a combination thereof. As a result, 20th Australian, 26th Australian and 3rd Indian Motor Brigades are narrowly cut off in Cyrenaica west of Tobruk, just like they themselves had cut off the Italians. With air attacks bearing down on them and no chance to esape, they surrender.
This leaves just the 24th Australian brigade defending Tobruk, soon to be reinforced by the rookies from the 18th. With only half as many forces available to the British and their overall commander captured, let's assume Tobruk falls no later than April 11th to the initial assault like it did later in OTL, with the large stocks of supplies and equipment found their captured intact (again like later in OTL).
Keep in mind that, at this time, Commonwealth forces were still in Greece, with the OTL evacuation starting only on April 21st (which barely succeeded with the ships only having to make the shorter return-trip to Crete).
What effects does this victory have? How far might Rommel advance under the circumstances? All the way to Alexandria's suburbs? Does Churchill still decide to try and hold Crete under the circumstances? Will he try and hold both Crete AND Egypt, allowing Crete to be defended only by local Greek and British units plus whatever escapes from the mainland, with the major formations transferred (without their heavy equipment) to Egypt?
Lastly, if the British still withdraw to Crete, might Hitler cancel Op. Merkur (airborne landing at Crete) and use the JU-52s to drop the paras in Egypt and carry out re-supply just as Rommel's tanks (with their infantry and artillery support left behind to facilitate the advance) reach the outskirts of Alexandria?