Would this work to reduce turbine engine spool times?
Short answer: no, none of what you describe is possible with current jet engines. You need more fuel because the angle of attack for your prop is increased, not the other way round.

Let's first go into some (simplified) detail how the turboprop works. A turboprop is basically a gas turbine engine with a propeller stuck on the axle. The propeller is the bit that converts the torque and rpm of the engine shaft into a forward thrust. The amount of forwards force is determined by RPM and blade pitch. This means that, even though you're at high RPM, you can 'feather' your propeller so that it doesn't produce any forward thrust (blades flat). At the same RPM, you can just put your blades so that they produce a lot of thrust. Of course, this comes at the cost of increased drag, so the engine has to 'work harder' to maintain RPM. This, in turn, is achieved by injecting more fuel.

Now, to a turbojet. For a turbojet, the thing that produces thrust is exhaust gas going backwards real fast. This is achieved by burning fuel so that the gas gets hot and expands, so that it wants to get out of there as fast as possible. For this, a jet engine needs a lot of air. It sucks in this air by a series of compressors at the front of the engine. Then, some fuel is injected, and after that there are some turbines that make sure the compressors can, and after that there's a nozzle that converts the leftover energy into backwards gas velocity. Now, we want to increase the thrust. This means that gas has to get out of the back faster. Since the nozzle is generally fixed, this means that you just have to get more air out of the back (just like a garden hose with the tap half open or full open). This means that we'll have to suck in some more air from the front, burn some more fuel to make sure the engine keeps spinning, and more air will come out of the back. 

Simply increasing the fuel flow doesn't work. Jet engines operate 'lean' (more air than necessary), and the extra fuel you suddenly inject will burn up with all the excess air, which might blow up your engine. If your engine doesn't blow up, you're left with too little air to burn all the fuel, and you'll get a lot of nasty things like soot, carbon monoxide and unburnt fuel coming out of your exhaust. This means you just inject some extra fuel, wait for the engines to 'spool up' so they suck in more air, and like that gradually increase your engine RPM and linked to that the gas velocity.

Two things remain: why use fixed nozzles? This is because physics dictate that, for engines to be efficient, the exhaust velocity needs to be as high as possible; in practice, the exhaust velocity will be as close as possible to the speed of sound in the hot flue gas. Using a variable nozzle would mean that your engine never works efficiently at half power. Furthermore, it adds complexity, weight and thus costs more.

Another thing: I discussed turbojets, not turbofans. In theory, one could have variable blade pitch on a jet engine. However, remember that the fan is not just for thrust (even though high-bypass engines are generally more efficient), but also to make sure enough air enters the compressors; 'feathering' them would mean that your engine might not get enough air. However, your engine would need a lot of air, because it needs to operate very lean to make sure it can handle the extra fuel flow for when you increase the blade pitch. Furthermore, a variable pitch on all blades would mean a lot of complex engineering, which would make the engine a lot heavier.

Long story short: one could design jet engines with variable pitch, but the above wall of text is only the beginning of why that is an engineering nightmare. Without variable blade pitch or nozzles.