For the Verse: C Fm Bb Fm Bb C
It´s clear that C is the tonic, but as you said early the rest of the chords seems to be taken from Cm intead of C. That´s a vague answer so let´s examinate it, shall we?
In C: I iv bVII iv bVII I
I iv it´s also V i, so let´s see what happends in terms of Fm.
In Fm: V i IV i IV V
This is now easier to look at, because we´re talking about the Melodic Minor scale (i ii +bIII IV V vi° vii°).
Despite the name, the scale has a Dorian sound with the difference of V instead of v; that´s why someone above recomended to use F Dorian.
But as we discussed already the tonic is C not Fm. Being C the fifth of F, that means the tonality used for the verse it´s the fifth mode from F Melodic Minor, also known as "Hindu scale", so the final answer is C Hindu (I ii° iii° iv v +bVI bVII).
Even thou we came to a satisfactory conclusion it is equality as right to use C Minor, F Dorian, Bb Lydian Dominant, etc.. (As long as you avoid dissonances like C against Cm). In practice whatever suits you it´s cool as long you and your partners can understand with each other.
Actually Melodic Minor is a scale used together with regular Minor scale (also called Natural Minor for no confussion). Knowing these we can arguee how it moves from C Minor to C Hindu, because it may be interpretate as if it goes from Cm to Fm.
Now about the Chorus: Bb C Fm Bb C.
We have the exact same chords as before so it should be the same as before.
The key to undertand this kind of complexier harmony is first: find out if there´s any cadence (V I; VII I; IV I) at the end of the progression, that should tell you were is the tonic; next the second is to look at what kind of relationships you have compared with the tonic, if you know the modal chords (II #iv° vii lydian; I v bVII mixolydian; i ii IV v vi° Dorian; i ii° v minor; i bII v° bvii Phrygian; i° bII biii bV bvii Locrian), then you can find what the progression "tastes like", for example here I saw "i IV" and I was "yup, Dorian sauce" but in Dorian the fifth is v not V, so work wasn´t over yet. Searching scales with the Dorian flavour (1 b3 5 6) I got Melodic Minor which again, despite the name it is really a Dorian scale.
Look: Dorian (1 2 b3 4 5 6 b7); Melodic Minor (1 2 b3 4 5 6 7); Natural Minor (1 2 b3 4 5 b6 b7) as you see Dorian and Melodic they are almost identical if weren´t by that b7 against 7. A better name could have been Dorian Leading Tone or maybe Harmonic Dorian, but the name is part of history now.
Probably a bit to extense and late, but I hope this helps anyone who reads it.