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Over the Christmas break I heard this programme on BBC radio:
Shine On You Crazy Diamond

David Gilmour discusses his famous four-note motif B♭ F G E but what wasn't discussed and I wished had been was the first three chords of the song, which are:

Gm G♭ B♭

In the first chord, the minor third of the chord is a B♭ note, which then serves as the major third of the subsequent G♭ major chord.

And I see that Gm is the relative minor of B♭ major. But what function does each chord serve? Is there some sort of cadential preparation going on? It sounds nice, but not like classical harmony to me.

The move from Gm to GbM is definitely not a standard “classical” progression, although it does start to happen more and more in the late Romantic era and in lots of Impressionist and other neo-tonal styles. It’s perhaps especially common in the triadic minimalism of composers like Philip Glass. I don’t think it’s super common in rock music, which might be part of why it sounds so fresh in the Pink Floyd example.

You’re exactly right that the “connective tissue” of the chord change is the Bb common tone. David Lewin called this move SLIDE, and neo-Riemannian theorists often call it P’ or “P prime,” because it’s the opposite of the connection they call P, which is short for parallel. P connects a major triad to its parallel minor counterpart and vice versa. This means the root and the fifth stay the same, and the third moves by a half step. P’ is the opposite, because, as you noted in your example, the third stays the same while both the root and fifth move by a half step. That’s why it’s the opposite of the parallel move.

Anyway, it’s an extremely cool sound, and it contributes a great deal to the moody effect of the beginning of “Shine on.”

EDIT TO ADD:
Sorry, I just realized you also asked about the move from GbM to BbM as well. This move also includes a common tone—notably, the same Bb common tone as the previous move—and half step movements in the other two voices just like in the P’ move. Different voices and directions are involved though. The move is called a “chromatic mediant,” and that’s the name for any two chords with roots a third apart and the same quality (major or minor). There are many questions on the site that explain chromatic mediant a if you want more details.

I had some trouble finding the part of the "song" you asked about. I assume it is this part starting at 4:59:

The tonality is G minor (I suppose G aeolian if you want to get picky.)

...what function does each chord serve?

If we are wearing out music theory hats, function means identifying the chords within a key and the harmonic roles of tonic, pre-dominant, or dominant.

This is how I labelled it...

    Gm         |Gb   Bb     |Eb dm cm Bb |F           |

Gm: i          |?    III    |VI v  iv III|VII         |

Right off the bat we see there isn't a dominant function. There isn't an actual V or viio. That isn't unusual in rock music. But it lets us know we are using a different playbook than that of classical harmony.

Of course the question is about the Gb major chord.

Technically it's non-functional, because it isn't a chord from the key or a closely related key.

As @PatMuchmore points out Gb is a chromatic mediant of Bb. You might also say it's a chromatic passing chord, because of the step-wise motion connecting the chords. Although the bass moving by roots sort of undermines the idea of a passing chord. Also, the chord is harmonically important so calling it 'passing' isn't really appropriate.

If Roman numeral symbols is the concern, there isn't one to give it. Either skip it, or just write Gb or 'chromatic mediant` under it.

It's worth noting that while many times a chromatic mediant chord is often a borrowed chord (like C and Eb in C major where the Eb is borrowed from C minor) in this case the Gb is not a borrowed chord.

Significantly, the Gb could be respelled F# in which case it becomes a non-functional chord built on the leading tone. In classical harmony that leading tone chord would be a diminished chord and strongly indicate the key. But that doesn't happen here. So while these chords on the whole are diatonic the progression eschews basic classical harmony.

Of course, calling Gb non-functional should not be misconstrued to mean 'bad', 'wrong', 'weak', 'unimportant', etc. This chord creates the whole psychedelic feel of the progression. Without it the progression would be plain diatonic.

Although Michael Curtis said otherwise, I would argue that this is a borrowed chord.

Basically the G♭ is used to give the move to from gm to its relative major B♭ the character of a proper modulation like you'd find in classical pieces. If the progression were just something like

gm     E♭ - B♭

ⅰ(/gm) | Ⅵ - Ⅲ

or even

gm     F  - B♭

ⅰ(/gm) | Ⅶ - Ⅲ

then the B♭ would sound just like a Ⅲ with a clear expectation of dropping right back into gm. If we instead make it

gm     F₇   - B♭

then this changes somewhat: the F₇ is a more forcing dominant, and now it sounds more like

ⅰ(/gm) | Ⅴ(/B♭)₇ - Ⅰ(/B♭)

Thereby, the B♭ chord effects a more prominent “opening up”, really places us in a major mood instead of just tossing in a major sound into the minor key. But the Ⅴ₇ - Ⅰ resolution is a bit too much of a classical cliché and would feel rather out of place in this piece.

What Pink Floyd do instead is cleverer: seen from the target key B♭, a chord is burrowed from its parallel key, i.e.

gm      G♭    -  B♭

ⅰ(/gm) | ♭Ⅵ(/B♭) - Ⅰ(/B♭)

with the ♭Ⅵ burrowed from the key of b♭-minor, i.e. we could also write.

gm      G♭    -  B♭

ⅰ(/gm) | Ⅵ(/b♭m) - Ⅰ(/B♭)

This reinforces the opening-up character further, because if we're effectively coming from the key of b♭m then the B♭ major chord has something of a Picardy third effect. I could lean further out of the window and say this foreshadows the clear Picardy-third ending of Part Ⅸ of the piece, but that's probably a bit too much of a leap.

That B♭ is a common tone to all these chords is certainly a relevant observation too, but I don't feel this is really important for the progression. I don't intuitively hear any pedal-tone effect in this part of the piece.

"It sounds nice, but not like classical harmony to me."

And there we have it. There's a note in common, Bb. That's about all the 'theory' we're going to find to justify this progression (if we can call a non-functional series if chords a 'progression').

There's not really much more to say about the 'chromatic mediant' Gb to Bb either. There's a note in common. We've got a label for it because, yes, it sounds good and therefore people do it quite a lot.