The Monster Manual's introduction provides general rules for telepathy in the section on Languages. The relevant passage for your question is the first paragraph:
Telepathy is a magical ability that allows a monster to communicate mentally with another creature within a specified range. The contacted creature doesn’t need to share a language with the monster to communicate in this way with it, but it must be able to understand at least one language. A creature without telepathy can receive and respond to telepathic messages but can’t initiate or terminate a telepathic conversation.
The description of telepathy in the Aberrant Mind sorcerer's Telepathic Speech feature adds a few additional limitations, most notably the need to share a common language:
To understand each other, you each must speak mentally in a language the other knows.
If you use this feature to communicate telepathically with a creature that "understands Common but can't speak", nothing mechanically prevents that creature from communicating with you telepathically by thinking in Common.
However, depending on why the creature is mute, the DM might decide that despite being able to understanding spoken Common, the creature doesn't actually think in Common, or is at least very out of practice at doing so. For example, a human who was "raised by wolves" and only joined society in adulthood would have grown up without language, and so even after learning a language, they may still not use it in their thoughts. Since the Telepathic Speech feature is limited to communicating in words, if the creature cannot think in Common (or chooses not to do so), they may be unable to communicate with you. And even if the creature can think in Common, they may not realize that they must do so in order to communicate with you.
Just like if you speak English natively and then later learn French, that doesn't mean that you naturally think in French, and if a French-speaking telepath tried to communicate with you, you could easily forget to think in French back to them, especially if you aren't used to communicating telepathically. And depending on how long it's been since you last spoke French, you might find it difficult to compose replies in French even if you can still understand what the French telepath is thinking at you well enough. For a real life example, I have a friend who was able to understand her immigrant parents' native tongue for a number of years before she learned to speak it proficiently.
The point is, understanding a language that is spoken to you is not inherently equivalent to being able to speak or compose thoughts in that language. However, D&D 5e has no mechanics for such cases, because for the most part the mechanics of D&D do not concern themselves with the inner workings of a creature's mind, and in particular there are no rules for partial knowledge of a language. So if your DM determined that a creature who "understands Common but can't speak" is also unable to reply to telepathic communication in Common, that was a ruling made by the DM, not a game rule. Whether the ruling was made by carefully considering the creature's situation and the specific nature of their muteness, or whether it was just an on-the-fly patch to avoid letting you short-circuit their mystery with a 1st level at-will feature, I can't say. That's between you and your DM.
With other forms of telepathy that don't require a common language (such as the warlock's Awakened Mind feature), I can't think of any reason a creature who understands at least one language wouldn't be able to telepathically exchange ideas with you.