Based upon my limited understanding of the servo operation I designed a circuit to generate 50Hz pulses that can slowly go from 2ms to 1ms and back, as desired. It requires just one LM339/393 comparator chip and a few passive parts.
The LTspice simulation is shown below:
The LM339/393 comparator U1 is configured as a relaxation oscillator to generate a 50Hz clock.
This clock signal generates an exponential ramp by R3C3 (2ms time-constant) which goes to one input of comparator U2.
At the positive edge of the clock pulse, the comparator output goes high (through output resistor R6) and the ramp starts rising.
When the ramp voltage reaches the Ref voltage as determined by R4, R5, and R7, the comparator output goes low.
This pulse width is 2ms when switch S1 is open and 1ms when S1 is closed.
At the fall of the clock pulse, the ramp voltage is rapidly pulled back to 0V through D1 for the start of the next clock pulse.
The time it takes the Ref voltage to change from one level to the next (and thus the time for the pulse width to smoothly change) is determined by the R8C4 time constant. This can be made as long as desired.
The time constant is quite short for the simulation, so the change in pulse-width (when V(pb) goes high) can be readily seen in a reasonable plot time.
R7 and R8 set the desired pulse-widths. These can be pots for ease of pulse-width setting.
Also separate switches can be used in series with both R7 and R8 if desired.
As a side note, I'm curious if any other servo users might have use for such a circuit to slow the servo movement. (?)