You haven't touched the pitch trim, so the trimmed AoA stays the same.
If the aircraft is in a turn then:
a. it's not accelerating downwards. It's descending, but at a constant rate, to first order. Therefore the vertical component of all the forces matches the weight of the airplane
b. it's acclerating sideways (or it wouldn't be in a turn). Assuming the attitude has some semblance of normality, i.e. nose first and tail last, then most of the force causing the sideways acceleration comes from the wings too - a.k.a lift vector tilted over etc.
Therefore the total lift force generated by the wings is larger than in level flight (i.e. an increase in wing loading)
Since the AoA is the same (inherent pitch stability of an airplane yada yada yada) the extra lift must be caused by an increase in the airspeed. Lift equation and all that.
This is to first order and ignores:
1. Trim changes caused by adding or reducing power, neither of which is necessary to go from level flight to a descending spiral turn.
2. What Denker calls the "long tail effect" - that in a steeply banked small radius turn the elevator has increased angle of attack, and becomes more effective at supporting the tail (or less effective at pulling it down). The nose tends to drop more than it otherwise would, the trimmed angle of attack (of the main wing) decreases and the aircraft actually speeds up in a spiral dive. It gets further away from a stall.
If you want to look into item 2 in more detail, google for "stick free manoeuvre point" and compare it with the "stick-free neutral point". This is an interesting place to start:
http://www.flightlab.net/Flightlab.n...u%232BA152.pdf