It's the same as for other limited resources
It does not really make a difference what feature that provides a limited resource we are talking about:
As explained in this answer on sorcery points, as written simulacrum only states that you cannot regain spell slots. That and that the way to regain hit points is the expensive alchemical procedure are the limitations provided on what you can regain. Everything else is regained normally.
The rules as intended have been stated in a tweet by Jeremy Crawford:
A simulacrum is meant to lose efficacy over time, essentially running out of juice, until only at-will abilities remain.
The errata for this that he suggested in that same tweet have however not been carried out to date. It's up to your DM to decide if he wants to follow this guidance, or allow you to use the spell as written.
In my humble opinion (my Wizard has simulacrum) the spell is insanely powerful, essentially doubling all your spell slots and giving you three more attunement slots and doubling all your actions, so it is better in this case to follow intended use and let all other abilities that are not at will get used up.
P.S. The limitations state that the simulacrum
never increases its level or other abilities
Could that "never increases (...) other abilities" means it cannot regain any used features even rules-as-written? You could argue that once it has used up any limited-use feature, regaining it would make it more powerful again, and increase an ability.
If that was meant, why then call out that the simulacrum cannot regain spell slots? That would already be covered. One therefore has to read this limitation in context: it is mentioned together with not being able to increase in level. I think this expresses that the simulacrum cannot acquire entirely new spells or skills, or increase its proficiency bonus or ability scores or other level-dependent class features. So, unfortunately it seems a bit of a stretch to take this as a rule for blocking replenishing of any ability. (And at least it cannot have been seen like that by the designer either, or he would not have suggested the spell needs errata to actually say that).