The Army got the plans for the Whittle engine, which was actually running in England, and gave them to General Electric. The Army then contracted with Bell Aircraft, an experienced airplane manufacturer, to design a land-based jet fighter using two of those engines. The resulting P-59 was, by all accounts, a dog.
Westinghouse created its engine from scratch with no outside assistance due to the secrecy imposed. It had an axial-flow compressor, which was the future, instead of the centrifugal compressor used in the Whittle engine, a dead end from the standpoint of increasing thrust significantly. The original Yankee engine, only 19 inches in diameter, worked well enough to demonstrate that the configuration was sound. One was flown under an FG-1 Corsair at NATC Patuxent River beginning in January 1944.
After development and qualification, two flight-rated versions of the Yankee powered the Navy's first jet fighter, the McDonnell XFD-1 Phantom. It became the J30.
So Westinghouse scaled up the 19-inch diameter engine to 24 inches for more thrust. It was subsequently designated the J34. It powered the McDonnell F2D Banshee and the Douglas F3D Skyknight (which necessitated the change of the McDonnell designation from D to H; the original Phantom became the FH and the Banshee, F2H). And also the Vought F7U-1 Cutlass, with the addition of a government-furnished afterburner that was subsequently provided by Westinghouse.
The Navy needed still bigger, more capable jet fighters so it held a competition for an even more powerful engine. Picking Westinghouse to scale its engine up again as the J40 must have seemed like a no-brainer. Originally, the J40-WE-6 was to power the A3D Skywarrior and the J40-WE-8, which was essentially the -6 with a Westinghouse-developed afterburner added, was to provide the thrust for the Douglas F4D Skyray, the McDonnell F3H Demon, and the Grumman F10F Jaguar. See Paul Christiansen's comment below for the subsequent changes in dash number.
It's available on Amazon.
Fair warning: it's detailed, technical, and comprehensive even by my standards, a blow-by-blow description of the engine and its development. There is relatively limited discussion of the airplanes it powered and the difficulties it caused in their development. For that however, you can read my much less detailed and technical books, Naval Air Superiority and Strike from the Sea.
Unfortunately, Westinghouse, which had been so successful up until then, failed miserably with the new engine as well as the electronic fuel control for the engine and afterburners. Development problems delayed the availability of the Navy's new fighters and long-range jet bomber, first by putting Westinghouse well being behind schedule on deliveries of engines for flight test and then by it not being able to qualify the engine for production at the necessary thrust, requiring Douglas and McDonnell to substitute other engines for the J40.
Westinghouse was not fully responsible for the Cutlass program problems. Vought was in part at fault for the cancellation of F7U-1 production because it wound up significantly overweight; however, availability of a satisfactory afterburner for the proven J34 contributed to program delays and a lack of enthusiasm for the airplane. Vought and the Navy attempted to address the weight problem with the J46 that was based on the J34 and developed concurrently with the J40. It was almost as disappointing but the F7U-3 was produced and deployed, albeit with a J46 of lower thrust and higher fuel consumption than planned, insuring that the F7U-3 Cutlass would forever be known as "gutless" and be replaced by other fighters as soon as possible.
Although production and support of the J34 continued, Westinghouse eventually exited the aircraft engine business.
Where did Westinghouse go wrong? One theory is the early successes did not result in a problem-solving culture within the engine division. Another is that the company did not continue to invest in technology and innovation like P&W and General Electric did. For example, to improve compression ratio for better thrust to weight and lower specific fuel consumption without incurring compressor stalls, P&W developed the two-spool engine and General Electric, the variable inlet guide vane concept. Westinghouse just kept scaling up the basic design, which eventually proved inadequate to the task.