Regarding this matter - I am a work so cannot do much research - but didn't the disinterest shown in the UK allow several patents to lapse in the 30s and be snapped up by other parties including those in Germany? More interest shown in the 30s and this data might have been better guarded and potential be far more advanced that OTL
I don't know about "several" British patents; Whittle took out just one, which he did indeed allow to lapse for lack of some sum on the order of 30 pounds to renew it--no other agency was sufficiently interested yet to pay for it!
Regarding others "snapping" it up though...it would have made sense if the Germans simply stole Whittle's ideas, which after all would be in legal public domain once he let the patent lapse.
However, Whittle met with Ohain after the war, and became quite satisfied that the German innovator had not himself read or heard of Whittle's patent. (Whether other Germans involved in developing jet turbine engines may have heard of the British invention, I do not know, but anyway Whittle personally exonerated Ohain. Their shared story was that Britons and Germans hit on the same idea independently at roughly the same time, though Whittle clearly had priority in retrospect.
Meanwhile there are other players, farther to the east and farther to the west, to suggest might have "snapped" up Whittle's work but again, the evidence seems to point toward independent development.
My major source on all this is Bill Gunston's
The Development of Jet and Turbine Aero Engines, 3rd Edition (Patrick Stephens Limited, 2002).
To the east, Gunston at any rate gives much credit to the work of Soviet engineer Arkhip Mikhailovich Lyul'ka who appears to have worked independently of either the Briton or the German. Indisputably the USSR benefited from the practically free gift of the Nene design in postwar years, but it would be quite wrong to assume Soviet jet engines were completely foreign in origin, though of course they appropriated German work alongside the Nene. But had all foreign sources been denied them they were inventing their own engines. To the west, no Americans developed any working turbojet models until they were given them by the desperate British, but on the other hand airframe designers at Lockheed were so confident that some sort of working jet engine would be coming along shortly they already had designs for a jet plane (lacking an actual jet engine yet) in hand before this.
The idea of the turbojet was generally "in the air" as the Second World War approached, then, and the only snapping up done was done by the British government's poor relations with Whittle, and free gifts of specific designs to both USA and USSR.
The Germans in particular might have done well to copy Whittle instead of pursuing their own independent course, for one decision undertaken in Britain--to focus on centrifugal compressors rather than the theoretically more advanced and greater potential axial compressor--was of great practical value in the war years, since the state of the art of metallurgy and turbine fabrication had to advance considerably to realize the advantage of the latter over the former in practice. In practice, given WWII period metallurgy, the centrifugal compressor could hold its own against then-practical axials, and proved to be more robust and reliable. With Germans suffering from lack of reliable access to many strategic materials, it may have served them far better to focus on centrifugal compressors. They didn't, which is pretty good confirmation of Ohain's claim to have worked independently of Whittle.
In the context of a WI about a more rational Air Ministry, much if not all of Whittle's work might have been undertaken earlier, but as someone noted above, the cutting edge of performance would have to wait on a succession of inventions in materials engineering.