If you want to "cheat", owing to the popularity of Barry Manilow's 1978 song Copacabana there are plenty of scores such as
from which it is easier to analyze chord progression as the intermediary step.
I "cheated" but I did modify some chord symbols to align better with what I hear in the original recording up to the minute 2:15. For the purpose of harmonic function analysis I also simplify / modify the chord symbols based on the more prominent chord notes in the recording, ignoring the -5th, 7th, -9th, 9th, etc. especially when they are only found in the melody.
Chords
Intro:
Gm9 | Fm9 | Gm9 | Fm9 Gm9 |
Verse:
Fm7 Bb9 | Ebmaj7 | Am7 D | Gm |
Fm7 Bb9 | Ebmaj7 | Am7 D | Gm |
D7(-9) Gm | D7(-9) Gm | Cm D7 | Gm D7/f# Bb7/f |
Chorus:
Fm Bb | Ebmaj7 | Fm Bb | Ebmaj9 C9 |
Fm Bb | Csus4 C7 |
Fm Bb | Gm C | Fm | D7 |
Bridge (same as Intro) to repeat from the beginning
Etc.
Analysis
Most scores curiously use a key signature with 3 flats but apart from the Chorus the piece spends a lot more time centered in G minor (including the ending), so to the ear it sounded as if the key is G minor. I agree with the 2nd score above that the key signature should have been 2 flats.
I would say that the Verse goes back and forth between Eb major and G minor although largely in (and ends in) G minor. But the Chorus stays solidly in Eb major (confirming your perception that the Chorus begins with a 2-5-1 progression) until the abrupt D7 chord at the very end to modulate back to G minor. What may threw you off is the prominent C major chords at the end of lines 1 and 2 below, which is the dominant to the ii chord (Fm) and how the 3rd line seems to be in F minor (1-4-2-5-1) before the final D7 chord. But since the melody of the 3rd line stays in Eb major (thanks @JohnBelzaguy) the progression can also be conceived in Eb major as 2-5-3-6Maj-2.
For the Intro (in G minor), I wouldn't call it a "progression" since it's just Gm9 and Fm9 chords shifting back and forth, ending in Gm9.
Verse: mostly in G minor
2-5-1 in E-flat major followed by 2-5-1 in G minor.
2-5-1 in E-flat major followed by 2-5-1 in G minor.
5-1-5-1-4-5-1 in G minor ended with 3 accented chords Gm D7/f# and Bb7/f (a quarter note each) to make a quick modulation back to E-flat major preparing for the Chorus.
Chorus: in E-flat major
2-5-1-2-5-1-6Maj (C9)
2-5-6Maj (Csus4 then C7)
2-5-3-6Maj-2 ending with D7 (dominant of G minor) to modulate back to G minor.
Bridge (in G minor) before repeating the Verse + Chorus is the same as the Introduction.