This is an interesting one - I haven't seen it this way before. Let me try and have a stab at it.
A rate 1 turn is expressed as 3 degrees per second or 2 minutes to go around 360 degrees. We normally think that any increase in airspeed causes the rate of turn to decrease, but that's assuming you're holding a constant angle of bank. In order for the aircraft to finish a larger circle in the same 2 minutes, it would need to fly faster (assuming it has the power) but by increasing its angle of bank, it can still achieve a rate one turn in the outer circle (within the same 2 minutes). The result is : the rate of turn is not affected by the radius of turn due to the change in bank angle.
A minimum radius turn is not related to time at all - it's a measurement of spatial distance. The aircraft would slow down to just before the stall and keep on as much bank as it can. It is trying to make to smallest circular smoke trail in the sky but he can take as long as he likes to do that. Conclusion : radius of turn varies directly with (square of) airspeed but inversely with bank angle.