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I think this is no "what if" thread!
The Allied fighters that could catch the Do335 (and other high-speed Axis aircraft) were already in the works. Several new designs along with existing model upgrades would soon be available. The Allies didn't sit around waiting for a new Axis aircraft to popup, the were constantly testing, enhancing and developing as time went by.Americans already had answers to German Panzer IVH, Panther and Tiger tanks too yet getting those answers into production and transported to Europe required about two years. What makes you think American reply to a new German fighter aircraft would be any faster?
The Allies will need to wait for 1945 to have, in numbers, a plane capable for 470 mph and better.
I've never been much interested in things powered by giant hair dryers
Production would be cancelled for same reason Henschel had to cancel production of Me-410. Ostmark engine plant was not building 1,000 DB603 engines per month as planned.
1944, nobody, in his right mind would've canceled a plane that is faster than Merlin Mustang, while featuring a heavy punch needed to kill the heavies.
1944, nobody in his right mind would've delayed mass production of Me-262 either. Unfortunately neither aircraft is much use without engines.
If you want 1944 Germany to mass produce a tandem engine fighter aircraft it must be designed for DB605 engines. However if you are willing to use the smaller engine why not begin design work during 1937 (i.e. when Dornier patented the rear shaft arrangement)? You could have it in mass production during 1941 ILO the problem plagued Fw-190A / BMW801 engine.
Ostmark engine plant production data. Chart calls it "RLM Plant Austria".
U.S. Strategic Bombing Survey Aircraft Division Industry Report
haven't seen any evidence that this led to a cancellation of Me 410 production by Henschel.