The general answer goes something like this:
Level flight, at it's most basic involves a balance of 3 factors:
- How fast are you going, relative to the wind?
- What is the angle of attack (AOA) of your wings, relative to the wind?
- How dense is the air that you are flying through, based on altitude, temperature, humidity, etc.?
Speed and AOA are variables that you indirectly control, via throttle (by increasing thrust) and yoke (by using elevator to alter pitch) respectively. The thing is, changing one variable (speed, AOA, or density) requires that one or both of the others change to maintain equilibrium. Want to stay in level cruise at a given density altitude after lowering throttle? AOA must increase. If the AOA stays the same, the aircraft will begin to sink because the current density cannot provide enough lift at that combination of AOA and speed.
That explains some of the "Why". Now let's consider your scenario. You have taken throttle out of the equation. Thrust is fixed and cannot be changed. This means that for any change to AOA, density must provide the equilibrium. Unless you are already at your stall speed, a slight increase in AOA without lowering throttle will begin a climb at slower forward speed because this new combination of AOA and speed are producing excess lift at the current air density. You have traded forward speed for extra lifting force. Because the throttle and new pitch are now fixed, the airplane will simply climb at it's new speed and AOA until altitude and temperature lower the outside air density to the new equilibrium point. At this point, the aircraft will once again be configured for level flight at the new density altitude. Also, because the density will change gradually, so too will the rate of climb decrease gradually. It will probably take a good while to fully level out.
The "lift" topic always sparks arguments, because technically, lift can be generated in any direction, depending on the definition of lift you are considering. An aircraft moving through the air on its side, may still create some small amount of force laterally, which opposes gravity, even though the wings may be producing no force at all. A helicopter in hover is utilizing all of its accelerated airflow for to counteract gravity, but a change in rotor disc tilt divides that total force between lift (opposing gravity) and thrust (opposing drag). An aerobatic airplane hanging on its prop is utilizing accelerated airflow in much the same way. An aircraft flying upside down is creating lift that is simultaneously against the pull of gravity AND toward the bottom of the aircraft.
Additionally, Bernoulli's principles work equally well in any orientation. An asymmetrically cambered airfoil mounted vertically will still create an area of lower pressure on one side than on the other, and is still capable of altering angle of attack to adjust deflection. Is this lift? The simple physics definition would say "No" because it is not opposing gravity. An aerodynamic analysis of the airfoil would say "Yes", the airfoil is creating lift according to its orientation, which in this case, affects only yaw.
It would be nice if the entire world could agree that Lift is the sum of all forces that act in opposition to gravity, and Thrust is the sum of all forces that counteract drag - regardless of whether these forces are provided by airfoil or prop (which is still an array of airfoils!). Unfortunately, for now, clarification is often required when discussing this problem in polite company.
In closing, I would like you to consider this: in your scenario, the reason that your airplane slows down is because, much like the helicopter, your prop disc is being tilted, dividing its power between lifting and thrusting. The drag on the airplane remains, so the added burden of lift is what leeches away the speed. The wings, on the other hand, no longer bear the full burden of opposing gravity, so the lift that they are producing DOES become less, but only because they are now sharing that load with the prop.