Learn Hardcore Portuguese (Portugal): Depois de vocês terminarem o relatório, enviem-no ao diretor. - After you all finish the report, send it to the director.

It is neither very formal nor very informal; it’s neutral and used in most contexts.
The verb that follows vocês is conjugated in the 3rd person plural, which you see later in enviem.

Compare:

tu – singular “you” (informal, 1 person)
vocês – plural “you” (neutral, several people)
o senhor / a senhora / os senhores / as senhoras – more formal “you”

So the sentence is talking to a group of people, giving them an instruction.

Why is it depois de vocês terminarem and not depois de vocês terminar?

Because Portuguese has something called the personal infinitive (infinitivo pessoal), which changes according to the subject.

The infinitive of terminar is:

(eu) terminar
(tu) terminares
(ele/ela/você) terminar
(nós) terminarmos
(vós) terminardes
(eles/elas/vocês) terminarem

Since the subject is vocês, we use the vocês form of the infinitive: terminarem.

If we said depois de vocês terminar, it would sound wrong to a native speaker. The verb needs to agree with vocês in the personal infinitive: vocês terminarem.

Is terminarem a subjunctive form or an infinitive?

In this sentence, terminarem is a personal infinitive, not a subjunctive.

It’s confusing because for many verbs the personal infinitive and the future subjunctive look identical in some forms:

Personal infinitive of terminar: terminar, terminares, terminar, terminarmos, terminardes, terminarem
Future subjunctive of terminar: (quando) terminar, terminares, terminar, terminarmos, terminardes, terminarem

However:

The subjunctive cannot follow a preposition like de.
After depois de, Portuguese normally uses the infinitive (often the personal infinitive) to talk about an action that happens before or later than another action.

So in:

Depois de vocês terminarem o relatório…

terminarem is an infinitive (infinitivo pessoal), agreeing with vocês.

Why do we use depois de and not something like depois que?

In European Portuguese, the normal structure is:

depois de
- infinitive (Depois de terminarem o relatório…)

Using depois que is typical of Brazilian Portuguese, especially followed by a verb in the subjunctive:

(BR) Depois que vocês terminarem o relatório…

In Portugal, people might understand depois que, but it sounds Brazilian and not natural in European Portuguese. For EP, you should stick to:

Depois de
- infinitive (often personal infinitive)

What tense/mood is enviem and why is it used here?

Enviem is the present subjunctive form of enviar for eles/elas/vocês.

In Portuguese, the affirmative imperative for vocês is formed using the present subjunctive:

(vocês) enviem – “send!” (you all send)

So:

Enviem-no ao diretor. = “Send it to the director.” (addressing several people)

In short:

enviar – infinitive (“to send”)
(vocês) enviam – present indicative (“you send”)
(vocês) enviem – present subjunctive, used here as imperative (“send!”)

Why is the pronoun attached to the verb in enviem-no instead of coming before it?

In European Portuguese, in an affirmative command (imperative), object pronouns normally go after the verb and are joined with a hyphen:

Enviem-no ao diretor. – “Send it to the director.”
Façam-no agora. – “Do it now.”
Digam-lhe a verdade. – “Tell him/her the truth.”

If the sentence were negative, the pronoun would usually go before the verb:

Não o enviem ao diretor. – “Don’t send it to the director.”
Não lhe digam nada. – “Don’t tell him/her anything.”

So the position of the pronoun depends on:

Affirmative command → after the verb (enviem-no)
Negative command → before the verb (não o enviem)

Why is it enviem-no and not enviem-o?

The direct object pronoun for a masculine singular noun like o relatório is o (“it”).

However, Portuguese spelling rules say that when you attach o / a / os / as after a verb that ends in -m, -ão, or -õe, they change to no / na / nos / nas:

fazem + o → fazem-no
põem + o → põem-no
darão + o → darão-no
enviem + o → enviem-no

This reflects a phonetic change: the final m or nasal vowel plus o sound like a nasal “no”, so it is written no.

So:

enviem + o → enviem-no (not enviem-o)

What does -no refer to in enviem-no?

The -no in enviem-no stands for the pronoun o, which means “it” and here refers to o relatório (“the report”).

o relatório → masculine singular noun
Direct object pronoun for it: o
Because of the verb ending in -m, o becomes no when attached: enviem-no

So literally:

enviem-no ≈ “send it” (where “it” = the report)

Why is it ao diretor and not para o diretor?

Ao is the contraction of a + o:

a = “to”
o = “the” (masculine singular)
a + o → ao

Both enviar algo a alguém and enviar algo para alguém are possible, but in European Portuguese:

a is very common and often a bit more formal or written:
- Enviem-no ao diretor. – “Send it to the director.”
para is also correct and can be a little more colloquial:
- Enviem-no para o diretor.

The meaning is almost the same here. This sentence simply uses the slightly more formal / standard pattern: enviar algo a alguém → enviem-no ao diretor.

Can we omit vocês and just say Depois de terminarem o relatório, enviem-no ao diretor?

Yes, and that would sound very natural.

Portuguese often drops subject pronouns when the verb form already makes the subject clear.

Depois de terminarem o relatório, enviem-no ao diretor.

Here, terminarem and enviem are plural forms and clearly refer to vocês, even if vocês is not written.

Including vocês is fine, but not necessary:

With pronoun: Depois de vocês terminarem o relatório, enviem-no ao diretor.
Without pronoun: Depois de terminarem o relatório, enviem-no ao diretor. (more typical)

Is there a difference between Depois de vocês terminarem o relatório and Quando vocês terminarem o relatório?

Both can introduce an action that will happen after the report is finished, but there is a nuance:

Depois de vocês terminarem o relatório…
- Literally “After you finish the report…”
- Focuses more on the sequence: first finish, then do the next action.
Quando vocês terminarem o relatório…
- Literally “When you finish the report…”
- Focuses more on the moment in time when the finishing happens.

In practice, in many contexts they are almost interchangeable. In European Portuguese, both are natural:

Depois de terminarem o relatório, enviem-no ao diretor.
Quando terminarem o relatório, enviem-no ao diretor.

The version with depois de more strongly highlights the “after X, do Y” sequence.

Would Brazilians say this sentence in the same way?

A Brazilian speaker would understand the sentence, but everyday Brazilian Portuguese would often sound a bit different. For example:

Depois de vocês terminarem o relatório, mandem o relatório para o diretor.
Quando vocês terminarem o relatório, enviem o relatório ao diretor.

Differences you might notice in Brazil:

They often repeat the noun instead of using the post‑verbal clitic -no, especially in speech:
- mandem o relatório para o diretor rather than enviem-no ao diretor.
Proclitic pronouns (before the verb) are more common than enclitic ones after the verb in everyday speech, but in formal written Brazilian Portuguese, enviem-no is still correct.

However, your original sentence is a very good model of standard European Portuguese.