Learn English – A situation when two or more people speak at the same time

If you are interested in a single word, consider babel:

A confused noise made by a number of voices.
‘the babel of voices on the road’
_{Oxford Living Dictionaries}

Not to be confused with babble, although it can be used as a synonym.

Hearing only babel, the teacher could not understand them.
Hearing only babble, the teacher could not understand them.

Your question is also open to phrases. The phrase talk over each other is most common in situations where more than one person are trying to make their point to either each other or to the same listener, but are not letting each other finish before speaking themselves. However, it seems to be a close match to the situations you posit.

They were talking over each other, and the teacher could not understand them.

However, the word simultaneously is a good single word to replace the phrase at the same time.

While perhaps more commonly used, babble is not specific to multiple people, and connotes rapidity in speech which is not part of your description. This difference in precision is likely unimportant in your example usage, since the usage context should sufficiently make clear the intended meaning.

The mass noun definition of babble makes it a near synonym to babel:

The sound of people talking simultaneously.
‘the answers were difficult to hear amid the babble of conversation’
_{Oxford Living Dictionaries}

However, babble has a singular form:

Foolish, excited, or confused talk.
‘her soft voice stopped his babble’
_{Oxford Living Dictionaries}

And its verb form specifically calls for rapid speech:

Talk rapidly and continuously in a foolish, excited, or incomprehensible way.
‘they babbled on about their holiday’
_{Oxford Living Dictionaries}

Babel is particularly appropriate in the case that there are multiple conversations being spoken simultaneously in more than one language, but the multi-lingual aspect of it is only to more closely match the Biblical metaphor.

⁵But the Lord came down to see the city and the tower the people were building. ⁶The Lord said, “If as one people speaking the same language they have begun to do this, then nothing they plan to do will be impossible for them. ⁷Come, let us go down and confuse their language so they will not understand each other.”

⁸So the Lord scattered them from there over all the earth, and they stopped building the city. ⁹That is why it was called Babel — because there the Lord confused the language of the whole world. From there the Lord scattered them over the face of the whole earth.
_{Genesis 11:5-9}

More about the myth of the Tower of Babel can be found on Wikipedia.

In common usage, it just refers to the incomprehensible noise of many people talking.

It is interesting to note that although babel and babble sound similar and have similar definitions, they are not etymologically related. Babel was taken from the Bible, and was a translation of the Hebrew name for a city that was referring to Babylon.
_{Origin section of ODO
Etymology section of Wikipedia
Etymonline}

On the other hand, babble seems to come to English from the German word babbelen, perhaps from Latin babulus or Greek barbaros, and probably originated as onomatopoeia of baby-talk.
_{Origin section of ODO
Etymonline}