Germany had the opportunity to field a true Jet-powered fighter well before the Me262 in the form of the He280, which was powered by centrifugal flow jet engines.
The HeS8 produced enough power to enable the He280 to perform well, though the fighter was intended to have a pair of HeS30 engines.
However, the RLM, once again, screwed itself by interfering with development and ordered Heinkel to develop the HeS011 instead.
Adding to the RLM's idiocy, the HeS30 development, which held a great deal of promise, was set back by infighting and eventual cancellation due to Schelp's favoring the Axials of BMW and Junkers.
*had* the RLM backed the HeS engine development (HeS3/HeS8/HeS30) early on AND not interfered, it is more than likely the HeS30 would have been ready for production sooner than late 1942 and the He280 would have been in the skies two years before the Me262.