But the ealy jets lacked the range for deep penetration. Even the P-80A (the longest ranged of the early jets) had a maximum range of under 1,500 mi. (probably ~600 mi combat radius)
The F-84E made it to 2,000 mi (with 2x tip tanks + 2x under wing drop tanks), but that wasn't until the very end of the 1940's.
But as SoD said, both sides could have had jets earlier, if Whittle's Power Jets had teamed up with Rolls in the first place the Meteor and the jet program would have been 2 years ahead in development. Or if more concentrated effort had been put on the Vampire's development it could have been in service quite a bit earlier.
Likewise Heinkel's projects could have been taken more seriously earlier on, and his class I engines should not have been canceled. (HeS-8/001, HeS-30/006) Or if some of the earlier designs like the HeS 6 had been developed a bit further and produced as an intrim measure. (the HeS-6 was already running well at 1,300 lbf thrust at the end of 1939) The He 280 could have been adapted to these engines inthe intrim for the other engines.
While inferior to the Me 262, the 280 could have entered service with necessary modifications with intrim powerplants by the end of 1942, at least for operational training. There were a few other bugs, mainly some problems with the control surfaces at high speeds (Mach limit .79) and problems with the tail structure, but minimal mosifications for operational standards: improving aileron design (not sure abot problems with this, but may have been similar to the P-47's), and using a single fin tail unit should have been sufficient. Range and weapons capacity weren't outstanding, but adequate at ~400 mi combat (improving with better engines, and some fuel capacity improvements) and the 3x MG 151/20 which were decent for a bomber killer and more than sufficint for a dogfight. (though it probably could have carried 2x Mk 108 ath the very least, maybe 3x 108 or even 2x larger MK-103 high velocity cannon)
The Soviet and US jet programs are a different matter though, as the Soviets got strted with government backing in the late 1930's (privately in the late 20's) but progress was modest, and much hindered by the German invasion. (political problems aside)
The NACA was doing a stusy on various jet and rocket schems starting in the mid 30's lookin into both RATO/JATO and primary power plants. They came to the decision that the use of such s a prime mover was impractical and the study was canceled ~1939/40. Studies on JATO/RATO continued. Then the British developments came in and in 1941 the NACA reintroduced a study on jet development (including their own pet motorjet/ducted fan project) but the only one fruit came from this: the Westinghouse J30. (the Firs indigenous US engine to run, and fly -on the XFD/FH Phantom-)
The GE TG-180 and TG-100 turbojet and turboprop projects which would become the J35 were started independantly by GE, parallel to the Whittle based I-A, I-14, I-16, and J31 engines. (also the I-20, cancelled) And of course the further development of the I-40 becoming the J33.
And the convoluted development of the P-59 was just sad, while it still holds an important place in history, it could have been much better. The same basic design, but with decent wind tunnel testing (partiularly the intakes and nacelles, but also the wing and tail) and less conservative design approach could have resulted in a decent combat quality aircraft. Though still inferior to the P-80.