The original Whittle engine was developed with private funds.

From "The Development Of Jet And Turbine Aero Engines" by Gunston:

pg 123: of which £200 came from an old lady who ran a corner shop near Whittle's parents in Coventry

pg 123: But a direct request to Air Ministry for a research contract in October 1936 brought flat rejection,

pg 124: Whittle could see that the only possible way to proceed was to take the gigantic gamble of running a complete engine.

pg 125: Indeed, there was little money for anything. While the RAF backed Whittle in every way they could - for example, by not requiring him to take the usual examination for promotion to Squadron Leader - the Air Ministry contributed nothing to Power Jets until May 1938, and Whittle had to watch every penny. He nearly cracked under the strain, which in fact was to get worse for seven years, not because of the Problems in developing the engine, but from the suspicion and enmity with which he was regarded by officials and manufacturers, and by the outrageous behaviour of the Company picked by the Air Ministry to produce his engine.

(Whittle's engine first ran in 1937).

It isn't possible to untangle its development and claim one or the other.