I practice the black art of thread necromancy to make a few points that were never made, or at best obliquely alluded to, in the year this thread ran:
While it is not entirely implausible (see my last point!) that both Whittle and Ohain might have had the success of their engines butterflied away, I do think that the development of the turbojet was close to inevitable, barring the total collapse of technological civilization. In particular the type of turbojet Whittle settled on for practical early development, using a centrifugal compressor, was essentially a glorified turbosupercharger. By the time WWII started a number of planes already had turbosuperchargers incorporated as basic parts of their engines and in the course of the war it became standard in many types of plane. Some kind of supercharging was essential to achieve decent performance at high altitudes and turbosuperchargers inherently tend to become more effective as the air thins. To be sure, Whittle had to greatly improve on the compressor's pressure ratio, tolerance of heat (and still more in the turbine side of it of course!) and mass flow; relative to an off-the-shelf turbosupercharger his achievement was tremendous. But in concept it was basically the same thing.
Furthermore, the most advanced piston engines of the late war and post-war period actually extracted a significant amount of their output horsepower from the turbine of their superchargers--this was called IIRC a "compound engine." The basic concept of a turbine that extracts useful power from a gas flow was thus not only quite familiar in theory but realized in practice quite as a byproduct of piston engine development. To be sure that was in principle a variant on the "motorjet" concept of using a piston engine to drive the compressor and it was not used to generate a gas jet but rather to augment the prop torque.
But meanwhile airplane designers were also designing radiators or the airflows around radial (aircooled) engines to use the radiated heat to augment the flow and thus provide significant thrust when exhausted; again this was a byproduct.
It's pretty hard for me to believe that someone somewhere would not look into the possibility of using the turbine exclusively, bypassing the piston engine completely, and then integrating all this insight into how to use waste heat to aid the thrust into designing ducts to harness the heat directly into a jet.
Perhaps a decade might intervene; perhaps without a big war to force lots of ideas to be tried (including alternatives that failed but might initially look more attractive) it would take a while to get going, and meanwhile perhaps WWIII would end civilization, thus mooting the jet engine.
But once developed the jet proved very attractive in various forms. First it conquered the realm of high-performance military aviation despite its early drawbacks (mainly being very fuel-hungry--the earliest jets were also unreliable but their basic simplicity compared to piston engines allowed them to pull even and surpass the highly developed but basically tempermental pistons remarkably soon); then as they became more efficient and reliable, they took the world of civil aviation by storm because however loud an early 707 or Comet was to people on the ground in the neighborhood of the airport, for the passengers they were much quieter and the nasty vibration that made piston planes so wearying was largely eliminated. Their speed (which, true, wasn't dramatically faster than the best prop planes--but still, faster) was another plus and eventually with the development of turbofans coupled with much bigger airframes, they proved to be more fuel-efficient as well. Even before that, the airlines saw a dramatic improvement in the overall cost-effectiveness of their operations; not only did faster (and soon, bigger) planes mean more passenger-miles per year of operations, they saved a lot of time and money due to the turbine's simpler maintenance and longer time between overhauls--more of a year was spent carrying passengers and freight and less in the hangar.
A world without jets is a world where aviation is simply less well developed, across the board. Helicopters too were revolutionized by being able to replace the heavy and tempermental radial engines with turboshaft power plants. Even prop planes today, except at the most modest level of light planes, have gone over overwhelmingly to turboprops, because they are lighter, just about as efficient, more reliable, and quieter.
Now on the other hand--if one wants to take Whittle out of the running

eek: he's one of my heroes!) it is true that when he started testing his first model of experimental turbine engine, it started to spin out of control--so he cut the fuel and
it kept on accelerating after he'd cut it off completely! Turned out it was aspirating lubricant oil in its sump and burning it up quite happily.
I suppose there is a real chance he might have died right then and there, and with him perhaps all British efforts at developing the radical concept--at least until those Messerschimtt fighters started intercepting allied bombers and fighters over Germany.
Stopping Ohain would probably require a more determined ASB.
And in the end, perhaps long after the war, the idea was just too straightforward and too much in demand not to have been developed by somebody.