Let's suppose that in 1939 the Air Ministry decide that the various very high power piston engines, Bristol Centaurus, RR Vulture and Napier Sabre are proving to be more trouble than they are worth and put their faith in Dr Pye's recommendation of Sqn Leader Whittles new engine. Then they put Power Jets into the hands of Hives at RR instead of Rovers.
The use of Rover to develop the engine in OTL put back the entry into service of the engine by over eighteen months at least.
Then instead of getting Gloster, a company whose last design was a biplane (wrong)! to design the first jet powered aircraft for the RAF the best designers in the industry are commisioned to submit designs.
To quote from Furse (Freeman's biographer)
"He also included Frank Whittle's jet engine in a limited class of 'war winning devices', and devised arrangements for its development and production.
Whittle was determined not to allow his enterprise to be swamped by one of the established aero-engine companies, and had himself proposed collaboration between his firm Power Jets, and Rover. The extreme difficulty of agreeing formal terms of co-operation between the two companies, and their intransigence, obliged Freeman and Tedder to insist that they worked together .... "
Even, later when the industrial & aero-engineering capabilities of this arrangement were found to be lacking, Whittle was still resistant to the idea. Nevertheless R-R took over the Rover factories with Rover being compensated by having the R-R Meteor tank engine factory.
IMHO - unless Whittle was an engineer in an established aero-engine company, rather than a 'maverick' development is most unlikely to be dramatically earlier.
That is: the Air Ministry wouldn't have invested more or earlier - no money about, and its too theoritical.
And, Whittle wanted to be in control not swamped by a big company.
So, sure it could be tweaked by weeks, or months - but years - no.