The whole circuit can be made from one logic IC - 4001. Although the CMOS type was used, you can use 74HC series instead, but you may need to adjust some values.
The astable is formed by IC1a/b and is fixed at a frequency of 50Hz. This provides a square wave pulse of even mark/space ration at the output of IC1b.
These pulses are shortened by the pulse shaper formed by C2/R3/D1. This differentiates the pulses to produce brief spikes. The diode prevents the negative spikes, generated by C2 when the astable goes low, from damaging the NOR gate inputs.
The final section is the monostable. This is triggered every time the spike generator produces a high pulse. The output of IC1c goes low and since C3 is empty, then so does the input to IC1d, thus producing a high at the output, being the start of the PWM pulse. As C3 charges, the input to IC1d rises until it reaches the logic threshold for high and sends the PWM output low - being the end of the pulse. Again D2 clamps the excess voltage appearing across the input of the gate to +0.7v.
With the values shown, the range can be varied using VR1 from 0.75ms to 2.25ms. This should allow correct operation of most servos.
Blue trace = IC1d input
Red trace = PWM output
Neutral or mid-point setting