Interresting question, but as I looked for more details, I did not found relevant information about the rules beyond this.

Looking for a better explanation, I will give my opinion about this.

The rule expected for the use of en and y, in this case, is that those words replace the attribute of the verb, to avoid redundancy. Therefore “Elle met du temps” becomes “Elle en met”. The extreme case, if you did not mention “du temps” earlier: “Elle en met, du temps” in which the comma denotes a true separation between “Elle en met” and “du temps” as two clauses.

As a native speaker, I also say “elle en met du temps” and moreover “Elle en met du temps à rentrer du travail”. I guess that en refers to temps and is intended to replace it. But “un temps” or “du temps” still remains, so I guess “en mettre un temps” is another pleonasm in our language, such as “Au jour d’aujourd’hui” or “Incessament sous peu”. Pleonasm is a way to emphasize a part of the sentence (such as “Au jour d’aujourd’hui”, which underlines the current date of the fact), so “Elle en met un temps” might emphasize the extreme duration (or our impatience). But it also often disappoints your listener as it denotes a misunderstanding of the language: use it carefully :).

Edit : ‘Elle en passe du temps’

“Passer du temps” and “mettre du temps” may not be used in the same context.
“Mettre du temps” describes the time it costs to execute an action.
“Passer du temps” describes the business involved in your spare time (gardening, playing, etc).

There’s another verb : ‘prendre’. ‘Prendre du temps’ means almost the same as ‘mettre du temps’

Because à is not the only preposition possible there? In fact, for me, it’s not even the default one. I would complete that with pour.

En is generally a more common word than y, and the presence of a de would tend to “contaminate” the pronoun.

Note that Elle en met un temps ! is also not the preferred formulation. The normal one is elle en met du temp.

Elle met un /du temps

This en refers to a certain quantity (of time) that is not defined. It is the same en as when you say “J’en veux”.

Usually we would not have un temps on its own, but there would be an accompanying adjective, such as “elle en met un temps fou”. If no specification we’d rather use du temps.

Elle y met du temps

Now let’s have another situation: your friend is a very conscientious person and usually takes her time to do things. You could say:

Elle y met du temps / y passe beaucoup de temps mais le travail est impeccable.

Because in this case, indeed, you take into account the fact that it is mettre / passer du temps à. It refers to whatever the time is spent on.

I’ve been trying to make up a single sentence using both pronouns en and y in the same time spent context and but can’t come up with anything, can’t tell if it is impossible or just me not being inventive.