it appears to be a typical basic builders fan. It will have a nominal air throughput in the region of 80 cubic metres per hour. A basic one cost the builders about £10, but it's possible yours is the 12volt "safe" one that can be installed directly over a bath or shower even if within 2250mm of the floor ("zone 1" in the attached pic, coloured orange).
Is yours within the footprint of a bath or shower tray and within 2250mm of the floor? If so it reduces your choice.
As your existing one is a budget fan, it most likely has plastic bearings that are worn out, hence noisy (the other cause may be clumsy fitting so the rotating fan blades rub on the casing). There are quieter fans available, but not many 12volt ones. If yours is, there will be a transformer, probably in the roof space and (I hope) readily accessible through a hatch. It may however be mains voltage. An electrician can easily check with probes.
It's a bit sad you don't have access above the ceiling as there are much more powerful ducted fans, but they are bigger so wouldn't fit on the underside of the ceiling.
You can get better results by adjusting the fan timer to maximum (usually about 20 minutes) run-on, and it would be better if it came on every time you turn on the bathroom light, because this will cause the bathroom to be ventilated more frequently, without any conscious decision or effort to turn it on.
An extractor fan works better with the door and window closed because water vapour is lighter than air so will naturally rise towards the ceiling and extractor.
This is an example of a high-quality modern fan, very quiet, that would fit, but it is mains voltage so please measure you ceiling height, and confirm if the hole is directly above the shower.