I'm not aware of any restrictions on aircraft armament calibers, 20mm was already encounterable throughout Europe by the mid-30's.
As far as I knew the point of contention during the League of Nations talks was the US contravension of the battleships treaty which restricted displacements to 35,000(?) tons and 15" guns during the 20's and 30's.
However Italy, England, the US and secretly Japan and Germany had promptly ignored displacement restrictions. And Italy, the US and secretly Japan had promptly ignored 15" gun restrictions.
Cruisers were limited to 20,000(?) tons displacement and Germany had loudly and publicly ignored this, building heavier cruisers armed like battleships, inventing the "pocket battleship" and causing a controversy over prewar military buildup (the Luftwaffe and army were one thing, but messing with the navies of the world = naughty).
Germany was not allowed to build battleships but also did that anyway, a little more quietly.
The US armed their oversized battleships with 16" guns and the comedic part was, for all their displeasure over it, Japan was fitting 18" guns to their top secret project which was also the largest battleship ever built.
Small arms caliber restrictions applied to Germany under the Treaty of Versailles and related to military and domestic forces, restricting caliber to a 7.65mm (a police officer could not lawfully equip a 9mm for example as this is a military arm and anti vehicle sized weapons in the 12-13mm range was definitely a no-no). They were also unable to equip artillery or heavy weapons emplacements.
Aircraft armaments were restricted to the same 7.65mm caliber under the treaty although it is reasonable the local production of 7.92mm caliber machine guns (probably 7.65mm inside barrel diameter anyway), was no problem. 12.7mm and 20mm armament calibers used elsewhere in Europe were definitely out of the question.
Needless to say Germany was in direct contravension of the treaty by 1923 at least domestically, more officially as elements of the National Socialists became the integrated Nazis Party as they gradually took over government. Individual military production warehouses were telephoned and advised well in advance of an impending treaty inspection, tanks and artillery were then moved out into nearby fields and cars and trucks placed in production warehouses for that week.
SA troops were rotated around 2-3 year army conscriptions to get more than 2 million trained by the 30's, whilst still keeping a maximum allowable of 200,000 under the treaty (I think officially increased from the original 100,000 at Hindenberg's request to the treaty nations, my memory's getting vague here).
Other troops were also trained in the Soviet Union prior to the war, away from the eyes of the treaty nations and of course the Spanish Civil War gave them combat experience.
Although there was little ignorance about Germany's contravension of the treaty, the extent of it was not known until the mid-30's, hence the uproar at the blatant appearance of oversized and overgunned cruisers, the so called "pocket battleships."