Sea Lion had little chance of working. Everyone underestimated the complexity of a large scale amphibious invasion in 1940. The Germans considered Sea Lion to be like a river crossing on a broad front. The Americans were pushing for an invasion of Europe as early as 1942. The Allies had the luxury of learning the hard way through invasions of increasing scale from North Africa to Sicily to Italy before undertaking Overlord. Even then it barely worked out. Curiously, there was little liaison between the US forces in the Pacific and the invasion forces in Europe.
The Germans had no large scale amphibious experience, they didn't stand a chance.
You're asking three big questions. I'm going to focus on one, what could be supplied from the air, because the answer renders the other two moot.
How strong were the British land forces available in Britain in the summer of 1940?
Here is the British order of battle 11 September, 1940, though that's just units on paper.
550,000 experienced soldiers were evacuated from France, but lost most of their heavy equipment. In addition, about 100,000 French soldiers were sent back to fight in France and were lost leaving about 450,000 in Britain. On top of that, Britain had a home guard of questionable use.
I'm not going to go into this much further, and you'll see why in a moment.
How large an additional force, Y, would be necessary to complete the occupation? Put another way, what was the necessary sum of X+Y to defeat the British army?
A lot. Here is the German order of battle for Operation Sealion. I count 28 divisions, on the scale of the 6th Army at Stalingrad.
They planned on 11 infantry divisions plus 2 airborne in the first wave, 8 Panzer and motorized divisions in the second wave, and six infantry divisions in the third.
Would this be enough? Its within the ballpark of what the Allies found necessary in Normandy, and the Germans were good at planning land campaigns. It's a big question and I'm not going to get into detail because the answer to the final question renders it all moot.
How large a force X, was necessary to establish and maintain a beachhead that could only be resupplied and reinforced by air?
Let's break that down into two questions. First...
How large a force could the Germans resupply and reinforce by air?
Germany failed to achieve air superiority over Britain in 1940 and never had a prayer of naval superiority. It's questionable how much of their invasion force would even reach the shore. Resupplying their beachhead is going to get ugly.
I'll use the attempt to resupply the 6th Army at Stalingrad as a yardstick. They were attempting to supply about 20 divisions totaling 210,000 men behind enemy lines while under air attack.
German strength in the pocket was about 210,000 according to strength breakdowns of the 20 field divisions (average size 9,000) and 100 battalion sized units of the Sixth Army on 19 November 1942.
They calculated they needed a minimum of 300 to 700 tons per day. That is to maintain a defensive perimeter with enough fuel for counter attacks. Maintaining offensive operations would require far more, I'm going to guess 1000 tons.
By pressing every available aircraft into transport service, the Luftwaffe was able to transport about 85 tons per day. The best they ever achieved on a single day was about 250 tons. They lost about 500 aircraft in the operation, half their total committed capacity. The 6th Army starved.
The Luftwaffe was able to deliver an average of 85 t (94 short tons) of supplies per day out of an air transport capacity of 106 t (117 short tons) per day. The most successful day, 19 December, delivered 262 t (289 short tons) of supplies in 154 flights.
Let's get some rough numbers. 1000 Luftwaffe aircraft delivered 85 tons under fire or .085 tons per aircraft. 200,000 men require about 1000 tons of supplies per day for offensive action (rough guess), or 0.005 tons per man.
That means 1000 Luftwaffe aircraft, their best effort could supply about 17,000 men or about two divisions.
What about Crete?
The Battle of Crete lasted just 11 days over a relatively small area using 30,000 airborne and mountain light infantry. Easier to transport and supply using the 500 transports at hand. The Germans had air superiority and captured an airfield early. Even so, they lost 370 aircraft in just 11 days.
The British forces were about equal in number and were operating far from their home island and had the option to retreat and evacuate, which they did. Had they hung on another week, the German supply position and loss of transport aircraft would have become dire.
But 500 transports for 30,000 1940s light infantry is in the ballpark of the 1000 transports for 17,000 men of two 1942 mechanized divisions.
Now the second question.
Is this force large enough to hold a beachhead?
Not a chance.
Even with the dilapidated state of the British Army, two divisions did not stand a chance against half a million British. Remember, this is the German army of 1940 fighting with Panzer Is, IIs and a few III and IVs. Two divisions is not just the beachhead, it is the maximum invasion force they can sustain. All while steadily losing transport aircraft to the RAF.
In Overlord, the Allies barely held on with their initial eight divisions plus naval and air dominance. It took a build up of 39 divisions, a million troops, to break out of Normandy.
Even if the RAF was destroyed, there is no hope for Germany to supply an army large enough to defeat the British army and get off the beaches.