There were a handful of jet aircraft that flew before the Me 262. The first jet-powered aircraft, the Heinkel He 178, would fly in August 1939, a few days before the outbreak of WW2. Across the English Channel, the British had their own separate jet engine program going (using a centrifugal flow design rather than an axial-flow jet) that would fly its first aircraft - the Gloster E.28/39 - in May 1941.

While this seems alarmingly early considering how late in the war jets made their combat debut, these early jets had numerous problems that made them unworkable as anything but testbeds. Endurance and range was uselessly short for combat - less than an hour on the Gloster design - and the engines were unreliable and needed quite a bit of refining before being suitable for operational roles. Despite that, the Gloster E.28/39 was actually designed with provisions for machinegun armament (though none were ever fitted).

The first aircraft that could reasonably be considered a jet fighter would be the Heinkel He 280. Work on this began in 1939 once the He 178 validated the concept of a jet powered aircraft, and a glider prototype would fly in September 1940. Unfortunately for Heinkel, however, the engines intended to power the design ran into continual trouble (ultimately to be cancelled), and the entire program would be plagued by difficulties in development of the engines until it was ultimately cancelled in favor of Messerschmitt's design.

Messerschmitt's Me 262 would similarly be hampered by engine development, flying for the first time in 1941 only by mounting a propeller engine in the nose to validate the airworthiness of the design. Engine problems would plague development and delay introduction, even forcing significant redesigns. The signature swept wing of the Me 262, for example, wasn't part of the original design, and was only incorporated to keep the center of gravity in the proper spot when it was found that the engines weighed more than expected.

Before the Me 262 would enter service as the first operational jet fighter, however, a handful of other jets would fly. America would fly its XP-59 prototype in October 1942, and Britain would fly two prototypes in 1943 - the Gloster Meteor and the De Havilland Vampire. The Meteor would enter service just a few months after the Me 262 - although it would kept away from frontline roles - but the Vampire was significant as the first truly viable single-engined jet fighter.

As for planned designs, that's an entirely different rabbit hole to go down - design studies existed among pretty much every major manufacturer for a wide variety of jet aircraft during WW2, often with unconventional layouts.