if it vents into the void, that is shockingly bad. It must not do that.
The ventilation rules say the fan must be on, but fans are usually fitted with an isolator to cut the power, while leaving the light on, while they are being cleaned, serviced or replaced. Sometimes the isolator is fitted on the wall above the door, outside the bathroom, where you would see someone interfering with it while you were working. A good one can be very quiet.
If you put that £10 Steeple one in the bin and replace it with a "silent" fan of better quality, it is normal to have them fitted with an overrun timer, so they keep running after you have finished your shower, while the room dries out. As you have no loft for a ducted fan, I would go for one fitted in the wall, close to the shower but not directly above a shower tray or bath unless you can mount it 2250mm above floor level (this is an electrical safety regulation) and close to the ceiling (this is because water vapour rises).
electrical.theiet.org/wiring-matters/53/-images/fig11a.jpg
Here is an example of a very quiet 100mm (4 inch) fan
www.tlc-direct.co.uk/Products/ADQT100T.html
and this brand has a good reputation
www.tlc-direct.co.uk/Products/SLSD100CRZ.html
Though if it is going in the wall, and you have frequent showers, a 125mm (5 inch) or 150mm (6 inch) can be more powerful and quieter (because it can run slower).
When choosing, look at the airflow in cu. m/hr. 80cu.m/hr is the lowest power. Look at the noise in db and choose a low number. Look to see that the motor has ball-bearings. An integral shutter reduces draughts but may open and close with a loud snap. Get a cowl vent for the outside of the wall, it keep rain out and does not rattle in the wind like the little flappy vents.
This is going to annoy the builders as they will have extra work and expense and perhaps they will want to blame you or cut corners.
Look out your plans, because they should have included an extractor from the start.