next time you redecorate, put insulated tile backer boards on the cold wall. They can be about an inch thick, a hard insulating foam slab with a bonded face to accept tile adhesive.
Thicker would be even better, but most bathrooms are tightly packed. An old house may have thick plaster, if you hacked it off you might gain nearly enough thickness for the insulation.
In that position you will be lucky to get a replacement fan to fit straight in, of more than 80 cu metres per hour capacity. However some of the new ones are very quiet and very low electricity consumption so you might leave it running permanently. for example Even this expensive one is not much more powerful. However your old one is sure to be clogged with dust, fluff and dity, and the motor bearings may be worn or gummy.
If looking at extractors, look at the cu metres per hour rating, and the noise measured in db. If you were able to fit a fan above the ceiling, for example in the loft, or out of sight, maybe in an airing cupboard, you could have a larger but much more powerful inline ducted fan, which is also very quiet. They are rather unsightly. The steam MUST be ducted outside the house, and not blown into the loft.
If the extractor is directly above a bath or shower tray footprint, the electrical safety regulations are much more stringent and expensive to meet, due to the water danger.
here are some examples. Surprisingly, the Expelair ones do not get good reviews.
I recommend a cowl vent on the outside to keep rain and wind out.