The spellcasting rules say:

A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.

I assumed the order didn't matter, and an unwritten, but logical deduction, is:

If you cast a spell with an action, you can no longer cast spells or cantrips with a bonus action on that turn.

However, a mate has argued with me, that if you first cast a spell with an Action, then you can cast another with a Bonus Action, since casting as an action does not prevent casting as a bonus, only the other way around.

I think it doesn't make sense, but in fact, the rules have some ambiguity. Does "You can’t cast another spell during the same turn" mean "you can no longer cast another spell", or "you can't have cast and you can no longer cast another spell"?

The Order doesn't Matter

Casting a non-cantrip spell as an Action prevents you from using your Bonus Action to cast any spell, and any spell cast as a Bonus Action prevents you from casting any non-Cantrip spell as an Action.

The reading "You can't cast another spell during the same turn" is pretty blunt. It doesn't just mean that you can "no longer" cast another spell, it means the conditions under which you may cast a spell as a Bonus Action require that you not cast any other non-Cantrip spells that turn. The text doesn't stipulate on the order that spells are cast; you shouldn't consider that either.

Note that this has strange implications for Eldritch Knights and their Action Surge ability: An Eldritch Knight is allowed to cast two non-cantrips as Actions using their Action Surge ability, but if they cast any kind of Bonus Action spell, then both their Actions, when using Action Surge, must be Cantrips. That might seem strange or counter-intuitive, and I'm not going to contest that feeling, but that is how the game is designed.

As a quick reference, this question has a table that describes the possible permutations of action + bonus action spells that can occur in a turn, and which permutations are legal. All of the rows are order-independent.

The restriction applies regardless of the order

You must use a Bonus Action on Your Turn to cast the spell, provided that you haven’t already taken a Bonus Action this turn. You can’t cast another spell during the same turn, except for a cantrip with a Casting Time of 1 action.

No matter when the bonus action spell is cast, you cannot have cast another spell except for a cantrip with a casting time of one action for the entire turn. This turn-based stipulation does indeed mean the restriction applies retroactively. It doesn't matter what order as long as the two spells are cast (or attempted to be cast) on the same turn, the restriction applies.

Jeremy Crawford has also clarified and agrees with this:

Q: How do bonus action spell rules go when a noncantrip is cast as a single action then a bonus action spell?

JC: If you cast a spell of 1st-level or higher with 1 action, you can't then cast a bonus action spell.

Note that things can get confusing when applying this rule and this question has a nice table showing all the legal and illegal combinations.

The prohibition is for the entire turn.

The wording restricts the entirety of your turn, not a particular timing sequence (emphasis mine):

A spell cast with a Bonus Action is especially swift. You must use a Bonus Action on Your Turn to cast the spell, provided that you haven’t already taken a Bonus Action this turn. You can’t cast another spell during the same turn, except for a cantrip with a Casting Time of 1 action.

On a turn during which you cast a spell with a bonus action, at all times that count as "during the same turn" you can't cast a spell with an action unless that spell is a cantrip. This does not enforce a restriction on the sequence of spells you cast. It enforces a restriction on your activities over the entire turn.

Another way to phrase it is that a turn in which you cast both a bonus action spell and a non-cantrip action spell is an invalid turn, which means you couldn't take such a turn, no matter the order of events.