Generically speaking, carrier jets in the Pacific war toward the end would be just the wrong platform across the board, without even getting into technicalities of who makes what, carrier design or any of that.
Generically, a jet is a fuel hog as the earliest replies noted--this is not due to poor design or primitive state of the art, but the basic physics of what we are trading off propellers driven by piston engines for. They have low thrust at takeoff and landing due to the same feature that gives them an advantage for high speed and climb--the rapid exhaust speed. (Note that nowadays we generally actually make turbofans instead of plain turbojets; these "gear down" the core exhaust speed by heavier turbine load in one way or another, to impart a lower overall airspeed on a higher mass flow, so now "jet" engines have a lot more thrust on the runway and much superior fuel economy. Indeed Whittle had ideas along the lines of approaches to turbofans in the wartime period, but these would be more complex and heavier and more prone to failure, unacceptably so in these early days). Now that same basic physics consideration gives the plain turbojet a real advantage, even in rather primitive form, in rate of climb and maximum speed at altitude. Performance in that envelope is where you would consider wanting a jet over the very highly developed sorts of piston propeller planes the USN for instance had in great abundance. If the Japanese were in a position to challenge US warplanes in such envelopes in naval battles, there would be some pressure on to figure out how to get a jet working off a carrier deck.
But for the Axis in general and Japan in particular, the longer the war went on the worse their logistics got, and their access to exotic metals as well as basic infrastructural abilities like keeping their factories running with basics like reliable power and so forth was never so great to begin with, even in 1942 when they were mopping the floor with their enemies. By 1944, Japan's production capacity was in pretty miserable shape due to the USN's increasing effectiveness at cutting off shipping to the Home Islands; then the island hopping campaign brought US bombers in range of the islands and their home infrastructure was being decimated violently on a regular basis.
Japanese designers were pretty good, but they lacked access to advanced materials and for that matter production of even routine designs was failing. On paper they had some pretty nifty plans for high altitude interceptors and such; I forget offhand if they had jet planes on blueprints. But they could hardly construct them however well designed! American bombers were not going to be met by jet interceptors; I have the impression they were not escorted at all. If they were, or had it seemed necessary, the right plane for the job would be an advanced piston propeller fighter.
Watt per watt, jet engines share an advantage with diesels that was in fact attractive to the Germans in particular--high performance piston engines depended on very specific and hard to get aviation gasoline with very high octane rating--American oil fields, notably those in Southern California, produced suitable grades of petroleum for the USA to make all the high octane avgas the US and Commonwealth forces needed. The Germans in contrast had very limited petroleum supplies, and one of the more successful aspects of the Allied bombing campaign was decimating their distribution networks. When Romania fell to the Soviets, their major source was cut off. Diesel engines, and jets, can consume a wide variety of combustable fluids--I believe there was even a serious attempt to develop a jet engine running on coal dust! Japan too might have benefited, though I think the alternatives the Germans had, including using seed oils, would have been in short supply in Japan, which has practically no domestic resources to speak of whatsoever--certainly not coal (for a feedstock for synthetic fuel production--burning coal dust was a really desperate and impractical idea!) Jets could burn gasoline but they generally don't (an exception was a design for the USN, so that the same fuel could be used in both piston and jet carrier planes--they gave that up pretty fast though). If it is all the same in terms of Joules of heat potential to be released, diesel/kerosene type fuels are denser and somewhat safer, being much less volatile, and withal basically cheaper in that a wider variety of sources can yield acceptable mixes.
But being unable to make jet engines due to supply constraints and poor conditions for manufacturing, Japan did not practically have the options the Germans did.
The fact that Germans deployed a number of jet models operationally and the Allies never sent their jets into direct combat with crewed Axis aircraft (Meteors were used to intercept V-1 "Buzz bombs") relates to another basic feature. Well developed jets on both sides mean your front line top of the line war planes must soon all become jets, but in the early state of the art of the early '40s, on top of the basic fact that turbojets require (and deliver, but not at low speeds efficiently) a lot more power hence consume fuel rapidly for that reason, the core combustion was a lot less efficient, being relatively low pressure compared to the fantastic compression ratios high-strung top of the line highly developed piston engines could get, so for a given power actually applied to moving air fast as opposed to just waste heat, one had to throw even more fuel at the problem. This is why later jets would be notably less piggy with fuel, even before the turbofan revolution. Getting even the mediocre pressure ratios they operated with in the wartime period already involved engineering with very rapid rates of revolution at very high temperatures, pushing the state of the art into territory scarcely dreamed up hitherto--to be sure these challenges would be much surpassed later. But this is where the relative abundance and ease and reliability of access to exotic metals mattered so much to the Allies, who had access to practically the whole world with U-boats only somewhat impeding it; the German resource situation was much tighter, and Japan's made Germany's look easy! For lack of good materials, German engine reliability suffered badly--not that it was so great even for the British and Americans, where engine failures could have spectacularly bad consequences too, but for German war planes to have jets at all was a roll of th dice for crews on every sortie. Yet the Germans used them anyway, and the Allies did not--why?
Because, aside from the desperate fuel situation, given the constraints on early jet engines, their rapid fuel consumption hence limited range and endurance, and the high odds that engine failures would ground a plane if lucky and more likely destroy it and its operating crews horribly, the best use of these early essays in the craft was point interception. The Germans were being bombed by long range high altitude bombers escorted by pretty hot fighters with range to match. A jet had a long runway run to take off due to low static thrust, but once airborne the ample power supplied meant rapid climb, and of course high speed and ceilings, all of which was advantageous in attempting to intercept intruding enemy aircraft.
This is basically to say then, that using early jet designs in practical combat was a mark of being on the defensive! If the tables were turned and it was Britain trying desperately to ward off German air raids, we can be sure the Meteors and Vampires would have been rushed into service, perhaps long before serious bugs in engine reliability were worked out to an acceptable level of failures. Attackers did not require the sorts of altitudes, rapid ascent and speed the defending jets took advantage of for their basic mission, only insofar as they had to tangle with these interceptors were they lacking. Against this they absolutely needed all the range and endurance they could get, to strike deep into enemy territory.
Excelling in first use of jets in combat then is something of a booby prize.
Now consider the situation of US carrier based planes. 1) takeoff and landing on a carrier deck for a jet is inherently more problematic than for a piston variable pitch propeller plane such as US industry had perfected and produced already in mass quantities. 2) using the same avgas as the auxiliary piston planes the carriers also operate is possible but throws away some of the advantage early jets offered and exacerbates the endurance problem. 3) once airborne the jets have very little endurance hence quite short legs. 4) superior aspects of performance, high speed at high altitude, are superfluous against an arrested and deteriorating Japanese operation--in addition to their poor manufacturing and supply conditions rendering quite good advanced designs in the works vaporware, and leaving already developed models that are losing whatever edge of superiority they once had years ago, Japan had poor policies of training and proficiency maintenance. Experienced Japanese pilots and maintenance crews were among the best in the world, but lacking either ability or inclination to rotate these experts back to Japan in the middle of the war to train up another generation of them, as they were decimated in combat, they had no good replacements. So the threat to be combated was deteriorating, and existing piston planes had plenty of advantage in terms of speed, ceiling, and durability versus typical Japanese planes flown and maintained by typical crews, and this advantage just grew month by month even without any state of art improvements--and in fact Allied pilots and crew chiefs were always on the lookout for such improvements, often anticipating the designers back home with field modifications.
The Alllies simply didn't need jets in operation in combat; by the time the Germans could manage to reliably deploy them they were already on the defensive, and indeed being on the defensive explains why they turned to jets as much as they did. If they would want any anywhere, it would be on the European front to tangle with German jets and rocket planes, not against poor Japan. The wide expanses of the Pacific would be exactly the wrong place to try to make jets work. Eventually as the state of the art evolved to the point that projected foes would be anticipated to use jets more and more routinely, it would become necessary for carrier planes to go over to jet propulsion too. But there was zero urgency to do this in 1944 or '45! And certainly not in the Pacific Theater!
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I do think it is kind of pathetic that the USA had no jet combat plane worthy of the name by the end of the war. Lockheed was the first firm to supply a combat-useful American design to US forces; this is no surprise when one considers that Lockheed designers had been anticipating the jet revolution for at least half a decade and had a beautiful design on blueprints in the early '40s--lacking however a suitable power plant! They just assumed someone would design something to do the job someday soon, and fleshed out what kind of plane to build around it.
Whittle emigrating and being taken up by some American firm--Lockheed is my preference for this obviously--is possibly what it would take, but first of all Whittle was a patriotic RAF officer, and would hardly abandon King and Country as WWII loomed. He did eventually emigrate to the USA in the 1960s but I can't see him cutting and running until after the German threat was taken care of.
And if we had either Whittle defying my judgement of his character or some American born person emerging with parallel passions and insights being funded and developing a decent jet engine design for Lockheed, or any other manufacturer of your choice, the first suitable war planes would be for the USAAF, based on long hardened shore runways, and like the Meteor and Vampire, quite unlikely to be deployed in actual combat--probably reserved to CONUS and maybe later British bases, perhaps if the runways were good enough to Pacific bomber base, as defensive interceptors, just in case anyone in the Axis came up with some game changing way of attacking the home bases. Their production and distribution and training for them would be low priority during the war--given the massive expenditures across the board might amount to quite a notable effort, but necessarily still, a mere footnote compared to the main focus on perfecting piston planes. Perhaps as with the eventual layout of the B-36 we might see some (gasoline burning) auxiliary jets on bombers to give them a surge of fancy maneuvering ability to evade interception, and maybe something similar on heavy high performance fighters like the Juggernaut. This would be a matter of making a suitably high thrust engine light enough to be worth hauling around.