Hit point recovery can only happen through the alchemical process
If the simulacrum is damaged, you can repair it in an alchemical laboratory, using rare herbs and minerals worth 100 gp per hit point it regains.
Repair not heal
First note that this does not say heal it says repair. Though treated as a normal creature it is still a being made of ice and snow. It is that reason that it likely has a special way to repair it.
Allowing other forms of healing means that the alchemical process would be useless
This statement must preclude other forms of regaining hit points otherwise it is meaningless. Even though it does not say so explicitly, the above method is clearly meant to be the only way to regain hit points for a simulacrum.
It does not say that potions or healing magic or rest doesn't work for example, but if they were possible then the expensive complicated process for healing would be meaningless and there would be no point in including it. Sleep especially makes this meaningless because it is free and available to every creature/class and allows for healing completely. Why would anybody ever pay 100gp per hit point if they could sleep or use any of the myriad of other methods for regaining hit points available? This logic also applies to traits or features that let you heal.
Simulacrum is more specific than Wild Magic; Wild Magic is no more specific than any other healing ability.
There is nothing that makes wild surges a more specific effect than any other healing ability. If the simulacrum restriction was not more specific than all healing spells and abilities would work on it, and as per the above, would thus make the restriction meaningless.
Take for example cure light wounds which says:
A creature you touch regains a number of hit points...
It uses the exact same language as the wild magic surge and thus if wild magic surge worked then cure light wounds or any other healing spell or magical ability would as well. And if all those things work then there is no point in even having the alternate healing method.
Thus, we must read the simulacrum restriction as being more specific otherwise it would be pointless.
Jeremy Crawford supports this logic as well:
To restore hit points to a simulacrum, you must use the costly alchemical procedure mentioned in the spell.
Which means Crawford also implicitly agrees that the simulacrum ability is more specific by extension.
Spell slots cannot be regained by any means
The same logic applies to the spell slot restriction. Simulacrum says:
The simulacrum lacks the ability to learn or become more powerful, so it [...] [can't] regain expended spell slots.
When it says that it can't regain spell slots, it means just that. It doesn't say can't regain spell slots except by this and this feature.
This is more specific than abilities that allow you to regain spell slots by default; otherwise the restriction wouldn't function at all. Whereas most things can recover spell slots, a simulacrum may not by any means. If it didn't override other methods of recovering spell slots then it would be pointless to even write it, thus it is by necessity, more specific than those abilities.
RAI again agrees
Jeremy Crawford again supports this interpretation here:
Simulacrum—the spell's creation can't gain levels or regain expended spell slots (RAW). The creation is meant to be unable to regain use of any of its features that it expends (RAI).
Thus, no hit points or spell slots can be recovered using wild magic.