You done a decent job, your right, the fan has a permanent live, a switch wire and a neutral so even if it’s not running and switched off at the light switch it’s still live at the fan (unless there is a fan isolator switch 3-pole which should kill it) it’s something to watch out for.I'm no expert..so if you're not sure get a man in!
But...mine is a timer type which is linked to the light switch and runs for a while after the light is switched off.
It is wired with three wires..live,live from light and neutral..it doesn't use the earth. I just took a photo of the wiring terminal as the new one had the terminals in a different order,easy as they're marked. I found drilling the wall the worst bit as unless you get the same model backplates never line up!
I fitted the plate,wired it then switched the breaker back on to see if it worked..all was good so cover on.
Like I said though...IF YOU CAN'T DO IT OR ARE UNSURE... LEAVE IT!!!
It’s the same with changing a ceiling light, lost count of the number of times someone has tried to fit their new light, just put all the reds and blacks together and hey ho, trips out. An unsleeved black switch wire doesn’t like being paired up with a neutral.