They simply pushed extant ballistic missile technology further than anyone else. The US and Russia saw no value in building something like the V2. The USAAF in project MX 774 Hiroc started in early 1945 was the US entry into large ballistic missiles. These were planned for use with nuclear weapons.
Interestingly, US engineers at Convair (Charles Bossart for example), at Aerojets, and Hughes (Raytheon today) rejected all of the German V2 engineering as crude and inefficient out-of-hand having a chance to examine it. Bossart rejected the conventional airframe with tanks fitted into it for one using the skin as the tank wall and pressurization to give rigidity. His design invention for the Hiroc is still used worldwide in ballistic missile design today. Aerojets rejected the use of graphite veins to steer the missile in favor of a swiveling nozzle, something the Germans tried and failed to get to work. Hughes rejected putting all of the guidance system on the missile, like the Germans had, in favor of using far better US radar and radio systems to guide it from the ground remotely. The Azusa guidance system was copied by the Soviet Union and widely used up through the 1980's.
So, the Germans had a bigger missile but it was really a poor design.
As for jets, the US and British both had them too. In fact the first squadron operational jet fighter in service was the US P-59. Yes, the 412th FG was a training group and never left the US but they had three squadrons of P-59 flying early in 1944. Their purpose was to learn how to maintain and use jets along with train pilots on them.
The US and Soviet Union, pre-war both had better liquid and solid fuel rocket engines than the Germans had. Once the war started, the Soviet program ended, focusing on just smaller solid fuel rockets that had a clear combat use. In the US, solid fuel rockets outdid those of the Germans but were given limited application. The US saw little value in liquid fuel engines.
In fact, with just a few exceptions, German technology in WW 2 was no better and often worse than Allied.