When and how did the names for aircrafts go from being nouns (Hurricane, Spitfire) to alphanumeric codes (F-16)?

As noted, the British (Spitfire, Hurricane) and American (F-16) air forces have different aircraft designation systems. If you're interested in the reverse of the question, "When did (British) aircraft names go from alphanumeric codes to nouns?", the answer is 1918.

During the First World War manufacturers all had their own naming schemes with a variety of letters and numbers representing different elements. To pick a few examples - the Handley-Page Type O, Royal Aircraft Factory R.E.8 (its eighth "Reconnaissance Experimental" type), Airco DH.9 (the ninth type Geoffrey de Havilland had designed for the company), Avro 504 (not the 504th Avro design, the series started with the Avro 500 for some reason), Vickers F.B.5 ("Fighting Biplane") etc. Many acquired unofficial nicknames like "Harry Tate", "Nine Ack" and "Gunbus"; Sopwith types in particular seemed to lack an official alphanumeric designation so were known as e.g. the Tabloid, Pup and most famously Camel (from the "hump" in front of its cockpit).

In 1918, with the formation of the Royal Air Force, the Air Ministry decided to standardise naming as the designer/manufacturer followed by a nickname. The nicknames were to follow a pattern, such as zoological names for fighters (with certain exceptions, e.g. birds of prey were already used by Rolls-Royce engines), geographical names for bombers, and at first initial letters were assigned based on manufacturer. This was fine for the firm of Vickers naming bombers geographically starting with "Vi" such as the Vimy and Virginia, but the firm of Boulton & Paul obviously struggled with zoological names starting with "Bo" - their first fighter design was the Boulton & Paul Bobolink. The naming system therefore evolved over the 1920s and 30s to be more practical (removing manufacturer-specific initials; alliteration would continue to be a theme, but not a rule), and inspiring. Being confronted with a Bobolink would hardly be a terrifying experience (except for a field of grain), the zoological names for fighters were dropped in favour of names "indicating speed, activity or aggressiveness" - the Gloster Grebe and Gamecock were succeeded by the Gauntlet and Gladiator. That's how the Spitfire and Hurricane got their names.

Over in the US the Army Air Service adopted the Mission Letter(s) - Number scheme, which also evolved over the 1920s and 30s to the familiar system in place for the Second World War - B (Bomber), C (Cargo), P (Pursuit, or fighter) etc. resulting in the the B-17, C-47, P-51 and such. The Navy had their own alphanumeric scheme that incorporated a letter for the manufacturer - the F4F was the fourth Fighter type built by Grumman (who got the manufacturer letter 'F'), the SBD was the first Scout Bomber built by Douglas. US aircraft also had common names, often bestowed by the manufacturer - B-17 Flying Fortress, F4F Wildcat etc., and of course aircraft tended to acquired a variety of unofficial nicknames from their operators (often less than flattering).

The Second World War resulted in an interesting overlap (and sometimes collision) between the systems with large numbers of US-built aircraft in RAF service, ordered before the war and later supplied via lend-lease. The RAF used their standard naming scheme with an appropriately American twist, so bombers got US town names (Douglas Boston, Martin Maryland and Baltimore, Consolidated Catalina). The aircraft that would become the P-51 was first ordered by the British, and their name for it (Mustang) would stick in the US as well. Where aircraft had established names they were generally retained; the B-17 and B-24 in RAF service were the Fortress and Liberator respectively, despite not being geographical, the P-38 Lightning and P-47 Thunderbolt appropriately indicated "speed, activity or aggressiveness" anyway. Most aircraft ended up with the same name in both air forces; it was briefly suggested that the Bell P-39 should be named Caribou in RAF service, but this was swifly dropped in favour of the manufacturer's Airacobra. US carrier aircraft were initially named along British lines, seabirds for fighters and fish for torpedo bombers - the F4F Wildcat became the Grumman Martlet, the TBF Avenger was the Grumman Tarpon, but in 1944 the American names were adopted instead. A few aircraft retained different names; the Curtiss P-36 Hawk and P-40 Warhawk were slightly tweaked to Mohawk for the former, Tomahawk and Kittyhawk for the latter in the UK, and the C-47 Skytrain, the military version of the DC-3, became the Douglas Dakota.