What is the relation between velocity and rate of pitch?
A constant speed pull-up is basically a circular motion, albeit in vertical plane:

$$q=\frac{(n_z-1)g}{V}$$

where $q$ is pitch rate, $V$ is true airspeed assuming no wind, $n_z$ is the load factor (lift to weight) and $g$ is gravitational acceleration.

The amount of load factor is limited by the airplane and its flight condition. As you can see, the faster the speed, the smaller is the obtainable pitch rate for a constant load factor.

Note: a constant speed pull-up is a theoretical idea that only approximates a pull-up at the very beginning, before the airspeed begins to decay. But the conclusion is valid.