My understanding is the Allies had no knowledge of the German jet engine program until it was leaked in 1943 by Fritz Kolbe and their own program was run independent of German efforts.
Not strictly speaking true; yes, little was known about the existence of German jets before 1943, but RAF photographic reconnaissance unveiled the existence of the Me 262, the '163, the V 1 and the V 2 in the spring of that year. The person largely responsible for identifying the new aircraft was one Constance Babington Smith:
Obituary: Constance Babington Smith | World news | The Guardian
Constance Babington Smith - Telegraph
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- who worked as a photographic interpreter at RAF Medmenham and was tasked with looking for anything unusual on the advice of one Frank Whittle. Babington Smith noticed scorch marks on the ground at Rechlin and from their pattern was able to determine that the new aircraft were twin engined. She was also responsible for identifying the Me 163 the same way (she wrote a book about her experiences called Evidence in Camera, which was the motto of the unit at Medmenham).
Before these aircraft were actually captured in images, the British were aware of the existence of such weapons from spies, POWs and other sources, but of course, little concrete evidence existed that detailed exactly what the Germans were up to and little credence was placed on reports of these things, although in mid to late 1943 based on the images and various unreliable sources, the British built a reasonably accurate picture of what the Germans had designed. In the summer of 1944, a crashed Me 262 wreck was investigated and experts made an assessment of its capabilities, which, along with their previous assessments proved surprisingly accurate.
According to the book British Intelligence in the Second World War Vol 3 Pt 1, at the end of 1939 a former Heinkel employee brought to British Intelligence's attention the existence of jet propulsion projects by that firm and in April 1940, the designation He 280 became known to the British, along with a calculated speed of around 495 mph by a jet powered aircraft. In May 1942 an He 280 was actually photographed by RAF PR aircraft at Rostock. None of this had any impact on Britain's own jet programme and it really wasn't until the existence of the '262 became known that an escalation of effort behind getting the Meteor into production sooner was undertaken.
It's worth taking this into consideration if we are going to examine any instances where '262 production was advanced by a year; it is almost certain that British Intelligence would have known about it along the same time frame before the type entered service - more jets on airfields means a higher likelihood of them being spotted from the air - approximately a year earlier, so basing what might happen on what actually did happen, we can argue that the Me 262 threat might not have been as sweeping as we might like to assume if they had entered service a year earlier.
In late 1943 - early 1944, the British had fears that the Germans might be able to field some 1,000 jet and rocket powered aircraft by early 1945, but by the end of the year it had softened its stance on the threat and began to disregard it when more intel on both types became known from combat experience and from examining crashed examples, as well as the absence in large numbers, of the kind that were ominously predicted - of both types. These fears of large numbers of jets led to the bombing of the Junkers factory that produced jet engines and of Rechlin by the Americans, so there was effort put in place to counter the threat, which did play a part in lessening the jets' impact.