Well I am going to start by assuming that 15 is a trimmed, hands off, 1G, AoA at your starting speed for the sake of this discussion.

The lift coefficient is equal to the AoA multiplied by the slope of the lift curve. So constant AoA means constant lift coefficient.

Since lift force is equal to the lift coefficient multiplied by the dynamic pressure (determined by density and velocity, .5*d*V^2) and the reference wing area, and the lift coefficient and wing area are constants, in this scenario lift is controlled by dynamic pressure.

In steady, level, unaccelerated flight Thrust is Equal to Drag. You talk about increasing the thrust. Now that thrust is greater than drag you begin to accelerate. As you accelerate the you are increasing your dynamic pressure, and thus your lift, raising the nose. How far does the nose raise?

steady state climb rate is found to be the following (thrust-drag)*true airspeed/weight. So you will keep pitching up until your climb rate lines up with that equation. The airspeed that comes out of that equation will be your climb or dive speed.

Another way to look at it is that at 15 AoA your thrust is adding to your lift (not really, but it is helping to oppose weight) at a value of Thrust * SIN(AoA). So if you remember the previous lift equation of L = CL * q(dynamic pressure) * S (wing area) you can add T*SIN(AoA) to it.

L = CL*q*S+T*SIN(AoA)

As we are moving between different trimmed 1G conditions we are moving between points where L is equal to the aircraft weight. When you increase T something else has to reduce to balance the equation since AoA, CL, and S are constant. That leaves q. As such you will see a decrease in airspeed until you have climbed to a point where the air density is lower. Same thing with reducing thrust.