Hi Olly,
The way I'd break this down, is to look at our chords in an E Major Key:
E F#m G#m A B7 C#m D#o E
The chords we know are from the key of E are E and C#m and the A - the 1 and the 6 and the 4.
While not being familiar with the piece, I would say the C would be a bVI - which is borrowing from the Parallel E Minor Scale.
Where this gets interesting is in the addition of the b7 in C (C7).
C7 would usually signal a change to the F major key, but as you see, no F. So why did they use it?
The next thing I'd do is look at the chords in terms of voice leading, maybe there is a common note they share, or a note is acting as a leading tone.
C E G Bb = C7
E G# B = E
Now if we look at it, not from a key standpoint, we see something interesting.
The B moves to Bb - 1/2 step back
The G# moves to G - 1/2 step back
The E stays where it is.
The C is the remaining note.
What if we took a C and superimposed the notes of an E major with it? This is where theory gets fun, is these what if's.
C E G# and B - I'd have a Cmaj7#5. One might look at the E to C7 as almost a rootless Cmaj7#5 to C7 move.
At any rate we've drawn a possible connection. We have 2 notes that move a half step back, while the others remain unchanged, thus I wouldn't call these substitutes more than I'd say creatively borrowed chords to imply movement.
In terms of substitutions, Olly, the chords that are subs usually share common tones with the chord being substituted. In the Key of C - C being the I chord, we'd see that Em and Am - the iii and the vi fit the bill as possible C subs. Lets see why.
C E G = C major
E G B = E minor (common tones E and G) You could say that using an Em implies a Cmaj7 without the C root, can you see why?
A C E = A minor (common tones C and E) It's can also be a C6, can you see why?
Another common substitution is known as a tritone sub, but thats usually related to Jazz and is built off the tritone of the V.
Now common variations, not subs, are usually chords borrowed from the Parallel Minor scale of a Key. Olly your example using Fm is a prefect example of that.
C Dm Em F G Am Bo C
Cm Do Eb Fm Gm Ab Bb
Notice in Cm, The F is Fm, notice also the presence of the Ab, this is where Cogs example is drawn from (That C to Ab7 or Amaj7)
In C, its not too uncommon to see the borrowed chords from its Parallel Minor.
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