The diagram you have is correct, but the terminals would usually be at the light fitting (or a junction box)
The fan is then supplied from the terminals makred SWITCH/FAN, which would usually be the isolator switch, usually in a 3 core and earth.
To use the timer delay, the fan needs to be supplied with a permanent live as well as the switched live (that comes on/off with the light). For a timed fan to work without the permanent live, it normally has to be have a short wire between the two terminals. Then it would work as you describe, only on and off with the light.
If you're checking the wiring at the fan, then the isolator switch *should* make it safe, but either test to be sure or isolate the circuit at the consumer unit before checking.
Is the isolator switch in the cupboard like this:
or like this:
If it's the second one, then the fan is probably only being supplied with the one live.
If it's in a cupboard then it's technically outside the zones, so it can be in the bathroom. If not in a cupboard then it can be in the bathroom outside of the zones, though traditionally is placed outside the bathroom door at high level.