With better funding (for machinists, draftsmen, fitters, engineers and test cells) Whittle could have flown a prototype two or three years earlier.
That would mean a few cannon-armed interceptors decimating German bombers during the Battle of Britain. Their greater speed would compensate for slow ground controllers.
Forget about dog-fighting because jet interceptors would be best at diving, slashing attacks.
Jet-powered night fighters would have to wait a couple more years as they developed more powerful engines .... powerful enough to loft two crew, plus radar plus 3 or 4 hours of fuel.
Later Jets would have sufficient range for the fast, low-altitude photo-recon missions flow by Spits and Mustangs. Their oblique cameras would fight German tanks hiding under trees.
Jet powered long-range bombers would need to wait until after WW2 because it would take many more years to develop jets powerful enough to lift bombers and fuel efficient enough for long missions.
I am always amused by Luft '46 sketches of German, jet-powered Amerika bombers because early jets guzzled far too much fuel to cross the Atlantic on their own. During the early Cold War, the USAF depended upon KC-97 tankers to extend the range of B-47s beyond America's borders.
Hopefully they'd be cannon-armed, the need for cannon was a recent lesson learned.
But I think where the impact of a handful of early jets (especially if this was their first appearance) would be strongest would be in "Messerschmitt Month", October 1940.
Imagine the situation - you're a Luftwaffe Bf109 pilot, flying at 30,000 feet carrying a 250kg bomb. You're up there hiding from what you've been told repeatedly are the "last 50 RAF fighters", too high to be bothered by Hurricanes and even Spitfires will struggle. Plus, you can cross the Channel and reach London in 17 mins and it takes a Mark1 Spit up to 27 mins to take off and reach this height. You're suffering a mild case of the bends in an unpressurized cockpit, and you suspect that a handful of barely-aimed 250kg bombs aren't gonna defeat England.....but you're on top of the World, King of the Skies, nothing can touch you without you seeing them laboriously climb to this height.
And then ....BANG.... cannon fire from above you, out of the Sun, as an early jet flashes past you and away at a speed you can't hope to reach.
OTL, the daytime raids of the Battle of Britain ended soon after this anyway, ATL, jet enthusiasts will be saying "It was the Jets what won it".