Let f(z) be the theta variable you are referring to in your question. Here are two parametric equations that should be very similar to what you have:
x(f(z)) = f(z)cos(f(z))
y(f(z)) = f(z)sin(f(z))
We can define the position p(f(z)) at f(z) as
p(f(z)) = [x(f(z)), y(f(z))]
The speed s(f(z)) at f(z) is the length of the derivative of p at f(z).
x'(f(z)) = f'(z)cos(f(z)) - f(z)f'(z)sin(f(z))
y'(f(z)) = f'(z)sin(f(z)) + f(z)f'(z)cos(f(z))
s(f(z)) = length(p'(f(z))) = length([x'(f(z)), y'(f(z))])
= length([f'(z)cos(f(z)) - f(z)f'(z)sin(f(z)), f'(z)sin(f(z)) + f(z)f'(z)cos(f(z))])
= sqrt([f'(z)cos(f(z))]2 + [f(z)f'(z)sin(f(z))]2 + [f'(z)sin(f(z))]2 + [f(z)f'(z)cos(f(z))]2)
= sqrt(f'(z) + [f(z)f'(z)]2)
If you want the speed s(f(z)) to be constant at C as z increases at a constant rate of 1, you need to solve this first-order nonlinear ordinary differential equation:
s(f(z)) = sqrt(f'(z) + [f(z)f'(z)]2) = C
http://www.wolframalpha.com/input/?i=sqrt%28f%27%28z%29+%2B+%5Bf%28z%29f%27%28z%29%5D%5E2%29+%3D+C
Solving this would give you a function theta = f(z) that you could use to compute theta as you keep increasing z. However, this differential equation has no closed form solution.
In other words, you'll have to make guesses at how much you should increase theta at each step, doing binary search on the delta to add to theta and line integrals over p(t) to evaluate how far each guess moves.