MOSFET Delay Provider
Use a simple AND gate and an RC delay.



When the input edge goes from 0 to 1 the capacitor input B goes HIGH immediately but C1 holds the A input LOW and starts to charge through R1. With B HIGH and A LOW the output, Q, is LOW. As the capacitor charges through R1 it eventually reaches a voltage level (50%) when the A input is seen as HIGH and the gate out switches HIGH. This takes Td = 0.7C1R1 (C in Farads, R1 in Ohms, Td in Seconds) approx. and produces a TURN ON delay at the leading edge.

When the input goes LOW the output of the gate follows it (after a small gate delay time) and goes LOW, even though the capacitor is now charged and holding the A input HIGH.  The LOW input allows the capacitor to discharge through R1 and it eventually goes LOW ready to start again.

If a faster discharge time is required a diode could be connected across R1 but this would not change the logic.