The subject of the OP was relating to ENGINES.
However, having said that, and since I am as susceptible to topic wander as the next person.
As to aircraft armament, I would try to bring back knowledge that would enable the manufacture of better and more compact Lead Computing Gun Sights (of whichever type). Aside from the obvious (greater hit %) it would make the use of gatling type weapons more practical.
One of the main problems with gatling guns (and other extremely high ROF weapons) is the lack of sufficient ammunition. For anti-bomber use, I am sure the Luftwaffe would have loved a 20mm Vulcan equivalent, if they could produce a platform capable of carrying it. As pointed out above the weight of the gun system and ammo is high, probably more than a single engine fighter could carry. It would have to be mounted on or near centerline, which would require synchronization (reducing the ROF enough to render the advantage moot), or fire through the hub. Maybe on twin-engined aircraft like the Me110 or FW187 (center fuselage) early-war, or something like the Do335 (hub?0 late-war. I do not think anything smaller could carry the system effectively.
Then there is the short fire time. This would be a problem due to both weight and volume. There is no way that a 20mm Vulcan system (600 rounds for a maximum fire time of 10 seconds at the normal LOW ROF of 3000 rpm) could be fitted into anything smaller than a Do335 size airframe. An approximate .50 cal//13mm system would obviously fit in a smaller airframe, but the high ROF/low fire time would be the same (or worse if max possible ROF of 8000(?) rpm was used). Again it would have to be centrally located, unless you want to cary 1 in each wing (ammo drums in the fuselage), in which case the airframe would have to be large (ie no Bf109/Spitfire/P-40/P-51/etc).
For bomber defensive installations (particular in the tail) the size/weight would not be to much of a problem (I think) but the low low fire time would be. I ran across a post-WWII US study on aircraft armament, and one of the problems mentioned relative to the B-17 and B-24 was that they would often run out of ammo for the guns, often before they reached the target. Can you imagine the increased chances of that with a 10 second time of fire instead of the normal 45-60 seconds. The major effect of bomber guns vs fighters was found to be the moral factor for the bomber crews and discouragement factor for the attacking fighter pilots. As to the adverse moral effect on the Luftwaffe fighter pilots, the Luftwaffe late-war reports and the post-war US study reported that the addition of the chin turret to the B-17 and nose turret to the B-24 only decreased head-on attacks by about 4% (the head-on percentage was still in the low 40% range, the tail attack in the mid-40% range, with other types accounting for the remainder).
A better Lead Computing Gun Sight would increase the hit chance for normal gun installations (whether in fighters or bombers) and would (maybe) allow the decreased time of fire for gatlings to be effective.