I can list a few examples of mentions of the English language in TOS. Since I am most familiar with TOS I will limit my examples and discussion of what language crewmembers speak and how they communicate with outsiders to TOS.
In "The Man Trap" The Salt vampire meets Uhura:
UHURA (to crewmen) The door to my quarters still rattles when it opens. Would you stop by and see if you can do something about it? Thanks, Bobby. (Green changes into African crewman) Crewman, do I know you?
CREWMAN: In a way, ma'am. You were just thinking of someone like me. I'm guessing of course, but you do look a little lonely.
UHURA: I see. So naturally, when I'm lonely I think of you.
CREWMAN: Ina cuvanea mwanamke turee.
UHURA: Una kafeeri Hur. You're Swahili?
SULU: He, whatever, was probably your crewman, too, Lieutenant.
UHURA: He must have been it. You know, I would have remembered a crewman like him.
And the audience would naturally assume that if the allegedly Swahili dialog actually was in Swahili the dialog spoken by the actors in English would have been spoken by the chracters in English. Though of course there is no proof that is the case.
The language the Starfleet members speak in "The Squire of Gothos" in not specified, but there are clues:
KIRK: Not yet. Put it on stun, not to kill, DeSalle.
TRELANE: DeSalle, did you say? Un vrai Francais?
DESALLE: My ancestry is French, yes.
TRELANE: Ah, monsieur. Vive la gloire. Vive Napoleon. You know, I admire your Napoleon very much.
KIRK: This is Mister DeSalle, our navigator. Doctor McCoy, our medical officer.
Mister Sulu, our helmsman, and Carl Jaeger, meteorologist.
TRELANE: Welcome, good physicianer and honourable sir. (bows low)
SULU: Is he kidding?
TRELANE: Und Offizier Jaeger, und der deutsche Soldat, nein? (gives a little Prussian salute then marches around) Eins, zwei, drei, vier. Gehen vir mit dem Schiessgewehr.
JAEGER: I'm a scientist, not a military man.
All other dialog in the episode is spoken in English by the actors. It is natural for the English speaking audience to assume that that the characters are speaking in English, but there is no proof. Certainly they are not speaking in French or German the rest of the time.
In "Arena", the Metrons transport Kirk to a planet to fight the Gorn Captain, saying:
METRON [OC]:You will be provided with a recording-translating device, in hopes that a chronicle of this contest will serve to dissuade others of your kind from entering our system, but you will not be permitted to communicate with your ship. You will each be totally alone.
And on the planet:
(Standing on rocks, silhouetted against a clear blue sky, Kirk makes an entry on his recorder for posterity.)
KIRK: This is Captain James Kirk of the Starship Enterprise. Who ever finds this please get it to Starfleet Command. I'm engaged in personal combat with a creature apparently called a Gorn.
(We see that the recording is also being transmitted and translated by the Gorn's device too)
(The bridge crew watch as enlightenment dawns and Kirk goes off in search of his other mineral discoveries.) GORN [OC]: Earthling! Captain!
KIRK [on viewscreen]: Who is this, the Metron?
GORN [OC]: This is your opponent, Earthling. I have heard every word you have said.
KIRK [on viewscreen]: All right. What do you want?
GORN [on viewscreen]: I'm weary of the chase. Wait for me. I shall be merciful and quick.
And they continue to talk for several more lines.
And as far as I can remember this is the only example in TOS where aliens are shown using a translator device to communicate, though of course some may use translator devices which are not mentioned or shown.
In "Space Seed" when Khan is briefly conscious on the Botany Bay:
(Kirk uses something metal in his hand to break the glass and open the compartment. The man is slid out on a trolley, gasping and awake. He tries to speak.)
KHAN: How long?
KIRK: How long have you been sleeping? Two centuries we estimate. Landing party to Enterprise. Come in.
Later, in the Enterprise sick bay:
(Khan sees a display of old medical instruments on the wall, takes a scalpel, and is back in bed when McCoy walks in to check on him. As the good doctor leans over, Khan grabs him by the throat and holds the blade to his neck.)
MCCOY: Well, either choke me or cut my throat. Make up your mind.
KHAN: English. I thought I dreamed hearing it. Where am I?
In "The Changeling" Nomad "erases" Lt. Uhura's memory, or at least messes it up badly. Considering how rapidly she is reeducated, I guess that she starts to remember as she is taught, instead of relearning everything from the beginning.
(Uhura is sitting up in bed, and reading from a monitor.)
UHURA: See the dog. See the dog,
CHAPEL: Sound it out.
UHURA: R, r, run!
CHAPEL: That's it, Uhura. That's very good. Now try the next one.
UHURA: The dog has a, The dog, (something)
CHAPEL: Not Swahili, Uhura. In English. The dog has a ball. See? B, ah, ll. Ball. Now you go ahead.
UHURA: (Swahili, then) The dog has a ball.
And again that indicates that the normal working language on the Enteprise is English.
In "Metamorphosis", when they meet Cochrane:
COCHRANE: You speak English. Earth people?
(Spock is working on the Universal Translator. It is basically a metal tube nearly a foot long, that can be held in the hand.)
COCHRANE: What's the theory behind this device?
KIRK: There are certain universal ideas and concepts common to all intelligent life. This device instantaneously compares the frequency of brainwave patterns, selects those ideas and concepts it recognises, and then provides the necessary grammar.
SPOCK: Then it translates its findings into English.
COCHRANE: You mean it speaks?
KIRK: With a voice or the approximation of whatever the creature is on the sending end. Not one hundred percent efficient, but nothing ever is. Ready, Mister Spock?
So the universal translator translates into English, which should thus be the main language on the Enterprise, and perhaps on all Federation Starfleet vessels.
I note that the UT looks a lot like the cylindrical communicator/translator devices the Metrons supplied to Kirk and the Gorn Captain in "Arena". Is it actually Kirk's device, did Starfleet reverse engineer Kirk's device, or is it just a coincidence that they looked similar?
"Metamorphosis" is the only TOS episode where their universal translator is mentioned. Universal translators in the subspace radio and the shipwide com system might be used other episodes, or maybe not.
It is possible that many of the Enterprise crew also speak a widespread language used for inter species communication, and that some of their communications with nonhumans on the Enterprise and off it is that language, as well as many aliens with previous Federation contact knowing and speaking English.
And it is possible that some advanced aliens have their own universal translators and use them.
In "Bread and Circuses" when they meet natives of the planet:
FLAVIUS [OC]: Don't move! Hands in the air!
SPOCK: Complete Earth parallel. The language here is English.
FLAVIUS: Are you trying to be funny?
SPOCK: Never. Colloquial twentieth-century English. An amazing parallel.
And no doubt there is a story why the people in this province of a Roman Empire on a distant planet speak mostly English instead of and/or along with Latin, but that is never mentioned in the broadcast episode, and perhaps never considered by the creators.
In "Spectre of the Gun" when they approach the Melkotian planet:
MELKOT [OC]: Aliens, you have encroached on the space of the Melkot. You will turn back immediately. This is the only warning you will receive.
SPOCK: Vulcan, Captain.
CHEKOV: It was Russian, sir. Every word.
UHURA: No, Captain. It was Swahili.
KIRK: Interesting. Telepathy.
SPOCK: Unquestionably. Most impressive
These examples show that the Enterprise crew have several different birth languages, but the workling language on the ship is English. So when actors portray crewmembers speaking English to each other, the characters probably usually are speaking English to each other.
Since TOS probably happens about 2 or 3 centuries after the 1960s according to clues in various episodes, the characters probably speak a future dialect of English which is translated in the 20th century English so the viewers can understand it better. And in some cases I suspect that the writers have transleted future English into 20th cnetury English rather badly and inaccurately. Especially when talking about scientific theories.
When actors portraying crewmembers and aliens are shown speaking English together, the characters may actually be speaking English with each other or some sort of very widespead space language translated into English for the sake of the audience. And possibly there are examples of both practices.
An alien translation device is shown being used in "Arena", and a Federation universal translator is shown being used in "Metamorphosis". Those might be the only two examples of the use of translation devices in TOS. Or possibly there are other uses of translation devices which are not mentioned or shown in ways that make their purpose obvious to the viewers.
And though the normal use of English as the working language on the Enterprise seems to be firmly established in TOS, methods of communication with outsiders, especially those encountered for the first time, is not explained so well, and it is uncertain what languages are used in many cases, o rhow often the universal translator is used.
And since the total number of Star Trek productions is about 10 times the number of TOS episodes, there could be a dozen times as many examples of evidence from other Star Trek productions as I have mentioend here.