While the early versions of Gloster Meteor flew with the engines derived directly from Frank Whittle's design, Messerschmitt did not invent engines that powered the Me-262.
Frank Whittle invented the jet engine in 1929. The RAF was unimpressed and turned his idea down. The jet engine was way ahead of its time. The fabric covered biplanes were still considered advanced for their time. The engineers were talking about metal monoplanes, but few of those actually flew, and the aircraft designers have not yet encountered the air compressibility and all the problems it presented.
Since the RAF was not interested Whittle kept the rights to his invention and, with a help of friends, turned to private industry to develop the idea further. Work progressed slowly through the 1930s due to lack of funding, but the first prototype, the W.U., run in April 1937. Despite a successful run of the prototype Whittle and his partners continued to struggle financially, but with the war clouds gathering the funding from the Air Ministry finally came through in 1939 allowing Whittle to build a flyable engine.
While Whittle struggled to attract the Air Ministry's attention (and funding) for his design, in Germany Hans von Ohain had better luck in getting government backing for his jet engine which he invented independently of Whittle. Though he started later than Whittle, von Ohain's prototype run on the same month as Whittle's and Heinkel He-178 became the first aircraft to fly under the turbojet power on the eve of World War 2, beating Whittle powered Gloster E.28/39 by nearly 2 years. Had the British Air Ministry and private investors been more forthcoming with funding, Whittle's design, most likely, would have taken to the air before von Ohain's.
The Gloster Meteor, powered Rolls Royce Welland jet engine, developed from Whittle's W2, became operational in July 1944, at about the same time as the German Me-262. Dr. Anselm Franz designed the Me-262's Junkers Jumo 004 engines. Unlike Whittle's and von Ohain's design where the air flow reversed itself twice while it passed through the engine, the air flowed through the Jumo 004 in a straight line.