In fact, the RLM funded the development of jet engines well compared to what air ministries in other countries did. Another quick translation quote from my old text
"Messerschmitt A.G. received the first information about the jet engine in 1937, and in the early autumn of 1938, the RLM, or the German Ministry of Aviation, convened a secret meeting with representatives of the aircraft and aircraft engine industry to give them a full report on the current stage of development of the jet engine. It was announced at the meeting that Messerschmitt A.G. and BMW A.G. will form "Southern Germany's development work concentration". In October 1938, Willy Messerschmitt gave instructions to his project office to investigate whether a single- or twin-engine jet should be aimed primarily at. The company received a formal jet fighter development contract from the RLM in late 1938. On January 4, 1939, the RLM issued a specification to the aviation industry called "Preliminary Technical Guidelines for a Fast Jet Fighter".
Messerschmitt A.G. after his preliminary studies started his jet fighter project, which eventually developed into the Me 262, on 1 April 1939. The company had settled on a twin-engine plane and presented its project proposal under the code P 1065 to the RLM on June 7, 1939. RLM approved the plan and gave permission to make the model..."
"…On the engine side, at the beginning of 1939, Brandenburgische Motorenwerke GmbH, better known by its abbreviation Bramo, received a development contract from RLM to develop a 600 kp jet engine. Its RLM project number was 109-003. BMW AG bought Bramo in July 1939 and also took over the development of this engine, at BMW it was initially named BMW P.3302 and later it was given the type name BMW 003. Initially, the intention was that Messerschmitt's P 1065 test machines would have been powered by the BMW P.3302s and on 1 Dec1939, a meeting was held between BMW, RLM and the Rechlin research station regarding the installation of the BMW P.3302 and P.3304 jet engines on the Messerchmitt A.G. P 1065 plan with fuel systems and gauges. However, these plans did not materialize due to the slow development of the engines, but after one unsuccessful test flight, the Me 262 was powered by Junkers' Jumo 004. RLM had ordered a jet engine from Junkers that could be quickly put into production in July 1939. The company chose an axial-charged solution, but otherwise decided to proceed cautiously in order to minimize the risks of the development work. [Kay 2007 p. 189, according to the same, development of 109-004 began in December 1939. According to Whitfield, in December 1939, it was decided to move directly to the construction of a full-size jet engine prototype when the scaled-down supercharger had broken during testing.] Since Junkers was simultaneously developing important piston engines for the Luftwaffe, the jet engine was initially developed by only a small group, which gradually grew as Junkers hired newly graduated diploma engineers. On the other hand, there were never any problems with funding, but there was enough of it and thus over time the group got good development and testing conditions. The design work started properly in October 1939. The first successful flight test under the fuselage of the Bf 110 took place on March 15, 1942. The successful tests prompted the RLM to order 80 Jumo 004 A test engines in the late summer of 1942 [according to Kay 2007 p. 190, 30 109-004 A-0 jet engines were produced]. 004 A was not intended as anything more than an experimental engine, the 004 B intended as a production engine would be a redesigned version that would be lighter and whose structure would clearly require less strategic raw materials and would be better suited for serial production. The first test bench use of Jumo 004 B took place in February 1943. The first Me 262 equipped with 004 B-0s was a V6, whose first flight took place on 17.10. 1943."
On He 280, the development of jet engines took time, the high temperatures in the combustion chamber(s) were a big problem, especially for the turbine blades, which at the same time had to withstand very high revolutions. The British also had difficulties in developing a durable enough metal alloy for turbine blades, although they did not have the same shortage of alloy metals as Germany. The biggest lack was of professionals with the right kind of technical training in all the major aviation powers. This is why the British concluded in 1944 that the war would be won with propeller planes, jet projects were developed with fewer resources just in case Germany made a surprising leap in jet engine technology and for the post-war period. The Germans were of course forced to try some radical solutions from the end of the summer of 1943 or make peace.
And the British did the same with the Meteor as the Germans did with the Me 262, first an order for a pre-production series, when the first test flights did not go as planned, the order quantity was reduced, when the test flights started to go well, the orders increased. One shouldn't throw good money after bad money, especially when there was a shortage of resources. Both had already learned that it was not worth producing large quantities of junk like, for example, Blackburn Botha and AW Albemarle or Me 210.
Even He 280 airframe was not ready in 1942. During the testing of the He 280, problems arose at higher flight speeds due to vibrations at the tail. In order to increase the airspeed to over 800 km/h and being capable of combat manoeuvres at those speeds, major changes to the airframe would have been necessary.
And what would have been done with it in 1942/43 in the main theatres of war on the Eastern Front and the Mediterranean. It was not suitable for frontline fields or the Sahara in general, and what would have been done on the eastern front with a fighter whose operating range at low altitude was something like 400 kilometres, practical operational time at low altitude probably closer to ½ hour.
And Germans did not foresee P-51B, they thought that they could handle USAAF heavy bombers with heavily armed fighters and zerströers.
On Hitler's influence
The Me 262 was originally designed to be able to carry bombs. When Göring asked Willy if the 262 could carry bombs, it was Willy who declared that of course 2 x 250 kg, even 2 x 500 kg bomb loads. For which Göring said, excellent, Hitler would have been satisfied with a couple of 70 kg bombs, but that is downright brilliant. Fulfilling Hitler's wish would have been easy, Willy himself was the one who ensured that bigger changes had to be made. Of course, when it became clear to Hitler in the spring of 1944 that all the completed 262s were pure interceptors, he had his famous fit of rage. In itself, Hitler's idea of jet fighter-bombers attacking the landing forces and landing craft at the critical stage of the landing was not bad at all.