"You and I" is translated Nous (i.e. "we"):

Nous avons fini notre travail.

… or, using on:

On a fini notre travail.

As an alternative to "we", you can say "You and me, we etc":

Toi et moi, nous avons etc.

Note that "Tu et j’avons" isn’t correct (i.e. I’ve never heard it).

I had a theory that you have to say "Toi et moi, nous avons" and that "Toi et moi avons" (without the "nous") is incorrect. The rest of this answer (below) will explore that theory.

If I search on linguee for ‘toi et moi avons‘ I see it’s used both with and without "nous", for example,

• Without:

  Toi et moi avons été surpris quelques fois par des choses que nous avons vues et pourtant nous croyions avoir tout vu.

• With:

  Et c’est vrai dans l’autre sens aussi, toi et moi nous avons besoin des décisions et des lois passées par les politiciens à Bruxelles.

My theory about grammar is that a word can be used as a subject of a verb if it’s a noun; and that "Le chat et le chien avons fini" is correct because "Le chat a fini" and "Le chien a fini" are both correct (i.e. "le chat" and "le chien" are nouns which can be used as subjects).

My theory continues, "Moi a fini" and "Toi a fini" are incorrect individually, i.e. therefore "moi" and "toi" cannot be used as subjects, and therefore "Moi et toi avons fini" is also incorrect.

This page Les pronoms toniques : moi, toi, lui… gives the following example:

On utilise les pronoms toniques :
-pour insister à l’oral :
Exemple : Moi, je suis française, mais lui, il est canadien et elles, elles sont belges.

However several francophones said that my answer (that "Toi et moi avons fini" is incorrect) is wrong; and I see it is used without "nous" on that page of linguee examples linked above: so perhaps "Toi et moi avons" is normal and is not incorrect; and/or perhaps it’s informal, familiar or "relâché" usage.

Or my my theory is just anglophone (i.e. not good French), based on this argument being used in English to show that "You and me have finished" is not as good as "You and I have finished".

Personal pronouns – nominative, accusative, dative, reflexive says,

The stressed forms of nominative personal pronouns are used alone:

Ee.g. Qui habite ici? – Moi. – Who lives here? – Me.

These stressed forms can come before the whole structure if the
pronoun is stressed: Moi, je travaille (It is me who am working).

This agrees with my recommending that you use "Toi et moi, nous".

On the other hand, Disjunctive pronoun says,

Disjunctive pronominal forms are typically found

• in a coordination with a noun phrase or another pronoun

  Mes parents et moi arrivons dans une heure.

  My parents and me are arriving in an hour.

I guess that, in the phrase "Toi et moi", each pronoun is being used ‘in a coordination with another pronoun’ (i.e. "moi" is being used with "toi" and vice versa) and the phrase "Toi et moi arrivons" is therefore correct (or at least not incorrect).

Toi et moi avons fini notre travail, mais pas eux. This is perfectly good french and quite usual.

The most common way to say this in everyday language would be

Toi et moi, on a fini….

More formally, you could say

Toi et moi, nous avons fini…

In writing, but much less likely in speech, you could say

Toi et moi avons…

The general principles at play are:

1. even though it is just as common in written French as it is in written English, spoken French doesn’t like to have verbs that aren’t immediately preceded by subject pronouns (hence the ubiquitous « ça c’est », « les mecs ils sont, » etc. etc.)

2. even though it is very common in written French, spoken French doesn’t often use nous conjugations, preferring on conjugations.

The reason « toi et moi avons » sounds stilted to native speakers is that it violates both these principles. However, it is “correct” prescriptively.

EDIT: I removed the phrase “extremely awkwardly” due to the comments. However, I maintain that « toi et moi avons » won’t be heard very often at all in spontaneous speech.

I agree with all previous answers but I just want to clarify this to use the following verb correctly. It’s like mathematics:

• Toi et moi = nous => avons
• Lui et toi = vous => avez
• Elle et lui = ils => ont
• Lui et moi = nous => avons

I hope it will help anyway 😉