As a general rule of thumb the weight of the of the projectile varies with the cube of the caliber. This assumes similar construction, materials and shape. A 13mm projectile will be roughly 4 times heavier than a 8mm projectile and a 20 mm projectile will be 4 times heavier than a 13mm projectile or 16 times heavier than an 8mm projectile. Simple theory gets thrown several curves (complications) in that the 8mm projectiles often have sizable components of lead in them while the larger ones do not. Steel having a rather lower density than lead. HE and pyrotechnical compounds have a much lower density than either. Wall thickness of the hollow projectile doesn't scale well. You need a certain minimum thickness for strength, both for centrifugal force (projectile spin) and to keep the pressure on the base of the projectile when firing from buckling the projectile in the barrel. This means, for practical purposes until the German mine shell, that projectile capacity did NOT change with the cube of the diameter but increased at a much faster rate. You don't need a much thicker shell wall on a 20mm projectile than you do on a 13-15mm projectile.
Yes, also all reasons a shorter, wider projectile would be more useful for chemical filler than a longer, smaller caliber one of similar mass. (space occupied by fuzing components would be part of that too)
On top of that, larger barrels have better wear properties, the drawbacks being heavier barrels and lower sectional density of the projectiles. (ammunition may or may not be heavier as well, that would depend more on the projectile weight, case weight -if any dimensions change, and propellant charge)
Something in the 14-17 mm range wouldn't be as good as 20 mm for HE/I but still considerably better than 12.7 mm, and even 13.2 mm would have some noticeable advantages over 12.7 mm. (larger caliber, shorter projectiles would probably be preferable on the whole -say around 17 mm with similar proportions to the Ho-5 or Shvak shells, though AP performance would diminish at range due to poorer ballistics even if velocity didn't drop)
a lot depends on timing, at certain times the Oerlikon guns required lubricated ammo (read greased or at least wax coated) which was frowned on for aircraft use. Grease/wax at 60 degrees F vs Grease/wax at minus 30-40 degrees let alone 60 degrees below. What is tolerated aboard ship/s is not tolerated as much in aircraft or on land. Please remember that while war was looming they weren't shooting just yet and what was desirable often held more importance than it did once the shooting started. Different countries may have had different standards of reliability.
Please remember too, that in the case of Great Britain, they were not buying just for use in the British isles but for use through out the Empire. ANY gun or ammo had to tolerate a wide range of climatic conditions.
Indeed, and there may be a better argument for some other countries adopting the oerlikon gun (plus cases of engine mounted guns would avoid some of the temperature related grease/wax issues).
The Oerlikon guns may have been more appealing in the US, especially for the USN given the typical low to medium altitude use and climates usually in play. (USAAC aircraft focusing mainly on coastal conditions would tend to fare better there too) Part of that is in hindsight given the awful trouble the hispano gave in US service, but the more practical pre-war advantages of lighter, lower recoil weapons intended for wing mounting would be significant as well. (especially with the USN more desperate for converting to 20 mm) The FFF may have been unattractive, but the FFL would be in a fairly reasonable performance range. (much better than the Army's M4 37 mm gun and similar to the ballistics of the 23 mm Madsen -slightly higher velocity but lower sectional density)
That said, larger caliber developments of the M2 Browning was probably the bigger missed opportunity. (also hindsight, but given the bureaucratic issues tied to US 'cannon' -guns above .60 cal- being manufactured with looser -artillery- tolerances, not only might the Oerlikon mechanism have fared better, but any light cannon staying at or below .60 cal would also avoid that problem and have tolerances akin to the .50)
There is a lot of talk about re-chambering and necking out (or down). Some of these guns had a few MM to spare as far as overall cartridge length goes and some did not. The designation, such as 23 X 106 does not tell you the over all length of the round, it tells you how long the cartridge case is without projectile. It also does not tell you haw fat the case is (diameter of the body of the case). It sometimes tells you how the gun was headspaced (how the cartridge is located in the chamber front to back) The 20mm Hispano case was 24.8mm in diameter and headspaced on the shoulder.
The Madsen 23 mm round was of similar length to the 20x120mm madsen round and I'm not sure on the exact dimensions, but I wasn't expecting it to directly fit into the Hispano, more suggesting that the hispano mechanism be adapted for a new 20 mm gun to replace the older madsen OR that the 20x110mm hispano ammunition be necked out to accommodate the 23 mm Madsen shell at some expense of velocity and possibly reduced propellant charge. (similar compromises to the MG-151 going from 15 to 20 mm)
But that assumes any country could secure a license for even the Madsen shells (particularly the high capacity HE/T rounds with self-destruct). It may have been just assumptions on continued import access to Denmark, but the lack of licensed production does imply the possibility of Madsen's terms being unfavorable or licenses not being offered at all. (granted, reverse engineering the weapons or ammunition could also have been a possibility, particularly after Denmark's capture)
In the case of the Browning, a 23 mm gun scaled up and engineered for the 23x106 mm round, or perhaps adapting the high powered US .60 cal anti-tank case instead would have been useful. (but that just goes back to the broader potential for the Americans to focus on the browning in general ... including scaling it up to fire that powerful .60 cal round in the first place -they went to the trouble of copying the MG-151, but didn't work on scaled up browning derivatives?)
Beyond that, the 'simple' cases of barrel replacements to otherwise similar guns would involve using identical maximum case diameters and base dimensions as well as projectile length/seating that maintained similar overall cartridge length. (so similar to the way the .50 BMG was adapted to 13.2 mm Hotchkiss and 14.3 mm in the .55" Boys)
The taper on the .50 cal Browning case is there for a reason, so is the shoulder. They can be changed but eliminating them or reducing them to mere vestiges of what they were is going to play havoc with the guns ability to function without some major development time.
I wasn't suggesting the taper be adjusted, though going much/any wider than the Boys did would mean a more significant change to the shoulder's position or dimensions. But wouldn't the 14.3 mm projectile of the Boys round itself be reasonable to adapt to the existing M2?
15~15.2 mm (60 cal) might have been doable, but the difference between neck and shoulder diameter would be even smaller and might cause problems. (likely widening the neck to 16.2~16.4 mm with the shoulder still at 18.1 mm, that's not much taper) A 17 mm or larger projectile would be a more definite problem and require repositioning of the shoulder and likely more modifications to the gun.
Looks like the Oerlikon FFF fired a 128 g shell at 600 m/s, the Ikaria MG FF fired the 134 g shell at same speed? The muzzle energy was 23-24 kj. For a 100 g projectile, that would be close to 700 m/s? The MG FFM used a bit smaller propellant charge, the 92 g shell went out at 700 m/s.
The MG FF didn't ended nearly as heavy as the FFL, but FFL received more development that gradually decreased it's weight from 43 to 34 kg, later to 30 kg. The development of the FFF/MG FF wasn't much pressed on, the only modification being the MG FFM, so the lower recoil could still operate the gun action. The FFL would be as easy/hard to install on the turets as MG 151 or Hispano V?
The lighter, developed form of the FFL combined with the lower recoil and shorter barrel should have made it more practical as a defensive weapon than the Hispano, though bulkier than the M2.
Since there is a violent opposition for the UK adoption of the 'big bang' Browning (

), any opinion on how well would it served in LW or USAF/USN?
As above, the US adopting the Boys (or even Hotchkiss) round would be interesting, but more so would be the powerful American .60 cal round. The latter wouldn't be rechambering though, but a complete scaling up of the weapon more akin to what the Japanese did, but likely heavier than the Ho-5 given that .60 cal was considerably more powerful, more powerful than the German 15 mm cartridge as well. (whether they kept it 60 cal or necked it out would be another matter ... or a matter of /when/ as well, given they did just that for the M39 revolver cannon much later)