Off-delay timer with only a relay and an on-delay timer
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Timing diagram and circuit schematic.

How it works:


TIMER1 is normally energised but when power is first switched on at the mains FAN2 will run until the timer times out.
When SW1 is closed FAN1 will run, RLY1 will be energised and power off TIMER closing its NC contact causing FAN2 to run.
When SW1 is opened FAN1 will switch off and RLY1 will drop out.
TIMER will begin timing and when the set time has elapsed the TIMER contact will switch to the NO position switching off FAN2.


This pre-supposes that both the relay and timer have changeover contacts (but you gave no details or datasheet links).