Updating with success story. I see a few other people have the same problem as me, so have added more detail about how I fixed it.
Ordered a side pull cord action switch, looks a bit like this:
Mine was rated for 240v 3A. Small little thing, about 1cm x 2cm. Took off the short cord and put on a longer cord. Switch has to be 2.25m or more above the bathroom floor, so you will need a longish cord, especially if you need it to be accessible to kids.
Took off the cover of my bathroom fan. Plenty of room under the cover. Baseplate already had a slot for a cord, but the cover didn't. A bit of work with a knife sorted that out.
Tested the switch with a battery and a LED. Switch activates once you pull the angled lever arm past roughly 45 degrees down from horizontal. However, the way it's manufactured, if you install the switch vertical, you're not guaranteed to get that >45 deg pull if you pull the cord straight down.
With mine, it looks like the cord is supposed to run up, over the top and down again. Idiotic - the cord is guaranteed to catch in the mechanism if you do that. The reason for that setup might be to ensure the wiring enters the switch body from underneath, to avoid condensation entering the switch body. However the mechanism also has a gap at the top so either way is just as bad.
I installed it with the cord hanging straight down, means the wiring enters it at the top, nothing I can do about that. There is a single screw fixing hole - won't prevent the switch from rotating and becoming useless when you pull the cord. I fixed that by supergluing the the pull switch to the fan baseplate. I superglued the switch at an angle, about 30deg from vertical so that any downward pull of the cord is guaranteed to pull the lever arm past 45 deg.
Wire connectors on the switch are the push-in type. Easy with solid core wire, an absolute pain if you are using stranded wire. My wires were stranded core (from switch to fan) so I used an unbent paper-clip to push in on the connectors and make some room for the stranded wire to go in. Wouldn't have gone in otherwise. Not a fan of the push-in connectors, but this switch only does momentary contact, so it should be OK. (I wouldn't use any latching (on/off) mains switch if it came with these connectors.)
Used 2 short wires, one from fan mains input to pull switch, and another one from pull switch to fan switched mains input connector. Both brown.
You may have a third wire coming into the fan labelled switched mains. Don't make the mistake I did and use that to connect to the pull switch. It's from the lights, and using it means the pull switch will only work when the bathroom lights are on.
All working now. Fan comes on for 10 minutes when cord is pulled. (Fan still self-activates when it detects humidity in the bathroom). Stinky bathrooms are now a thing of the past.