I was just about to ask this myself.
I have the same problem with achievements not working.
I've yet to see any specific advice (or even many complaints on achievements making me question if I was the only one) on how to fix this.
Making sure galaxy is not offline and similar were what little I could find from aging posts.
In the "my progress" tab I get "This game does not have achievements support." when I hover over "Achievements", which is somewhat odd with the list just below and the 4% average player achievement listed in the "Overview" tab.
My best hypothesis is that Cult sim is not talking to galaxy or talking in a way galaxy does not expect/understand, alternatively that achievements are disabled for some reason (on Cult sim or galaxy's side).
I don't think it's related to dlc as the steam achievements have had some players getting them from the latest Exile dlc.
Seeing as how the steam version has working achievements with a reasonable "completion curve" it's obvious that Cult sim has the ability to notice your progress and send that info.
With galaxy reporting an average achievement completion of 4% means it worked at some point, same with the average play time of 9h08m (even if Cult sim is a bit of a niche game this seems low to me).
So yeah, my thoughts on this, hoping this is explained/fixed.
Ultimately I find this to be a minor nuisance in a good game, now, if you'll excuse me, I must go and be a door to my congregation.