Soviet were developing several twin-engine fighters in 1939-41 but the crisis of 41 stopped most projects and only Pe-3 (the fighter version of Pe-2) went to production
Weren't most of them stopped because they were failures? Such as the Mig-5
Were they failures or did they fall out of favor due to shifting requirements? The MiG 5 seems to be the best of the lot and mainly held back by the cancellation of the AM-37 engines and possibly overall shift in emphasis towards low altitudes along with limited value of the later M-82 powered version. (an AM-38 powered low-alt heavy fighter/attack variant may have been more attractive, but there doesn't seem to have been interest in that and it would be directly competing with Il-2 production -and regardless of potential greater effectiveness and survivability of a higher performance fighter/bomber attack plane with more accurate centerline armament, the Soviets didn't seem to be favoring that direction of development)
However, with this scenario avoiding the Soviet evacuation and industrial relocation as well as having greater need for long-range, high altitude fighters as escorts, the MiG twin and its AM-37 engines might have been given much higher priority. A developed AM-37 would also benefit the Pe-8. Same for the AM-39 as a potential followon high altitude engine. (though the AM-35 may have been reasonable to apply to the MiG early on in the interim before production problems with the AM-37 were worked out, and while performance would obviously be lower, you'd at least save some weight with the lighter AM-35As)
I know the AM-37 was abandoned historically, and its existing problems may not have been totally solved without curtailing power output somewhat (at least in the short term), but given the success of the AM-38 (and later AM-42) and overheating apparently being the main problem on the AM-37 there seems to be plenty of room for solving those issues. The AM-37 seems to be to the AM-38 what the AM-39 was to the AM-42, and in both cases the low altitude versions seem to have gotten priority.
Beyond that, a smaller twin using Klimov engines might have been more attractive as an escort fighter. The Ta-3 seems more in that size class, though used rather bulky radial engines and the range cited on wiki isn't very promising. Still, it seems like an aircraft that would be better suited to the Klimov V12s than the large, fairly heavy 14 cylinder M-88 radials it used. (a bit like situation in the Fw 187 discussions regarding using captured Gnome Rhone 14N radials, or a bit worse even)
Fine points.
The AM 38 was indeed to the AM 35A what basically the Merlin VIII was to the Merlin III - an engine with lower S/C grearing, with most of everything else remaining the same. The difference was a bit lower compression ratio (7:1 down to 6.8:1), and the AM 38 was rated for extra 100 rpm more for take off. The S/C gear ratio for the AM 35A was 14.6:1, for the AM 38 it was 11.05:1.
BTW, the AM 38F was turning 2350 rpm for take off (and emergency; the impeller was of smaller diameter; CR was down to 6:1), 2150 was for the AM 38 and 2050 was for AM 35A.
The AM-38 really seems more related to AM 37 development given the timing, power output, weight, and structural reinforcements.
I'm sure an AM-35 derivative with supercharger configuration similar to the AM-38 would have been possible and allow for a substantial power increase at low altitudes, but not the 1,680 hp the AM38 was producing. (something in the 1,400-1,500 hp range seems more realistic, and should be a good deal lighter than the AM-38 as well, and somewhat lighter than the AM-35)
Can we at least agree that in terms of radar the Soviets aren't going to capture the later developments and will be stuck trying to make sense of whatever the Germans were working on, as most of the scientists would probably flee west? They were remain at a major disadvantage and maybe would be lucky to get some Wurzburg radars. By this point airborne radar has been developed, probably not even Hohentwiel due to the lack of need without an Atlantic front.
The most useful german technology captured would be examples and plans that actually fit better with the Soviet's more limited manufacturing capabilities. Some of the German turbine developments might be useful for both jet and turbocharger development, but a lot of that would probably be limited as well. If they captured Heinkel/Ohain's early development work and/or some of their materials that had been moved to museums in Berlin (I believe both the He 178 and some of their prototype engines were moved there) it may have been fairly significant. Using radial turbines in jets or especially turbochargers may have overcome some of the issues the soviets were dealing with. (at least those relating to turbine burnout due to thermal and aerodynamic stresses -radial turbines allow practical use of considerably lower quality metals and scale in size somewhat more easily than axial turbines -all modern automotive turbochargers use radial turbines)