In Volo's Guide to Monsters (pg. 81), the mind blade weapon is described:

Mind Blade

Weapon (any sword), rare (requires attunement by a specific individual)

Mind flayers can turn any nonmagical sword into a mind blade. Only one creature can attune to it: either a specific mind flayer or one of its thralls. In the hands of any other creature, the mind blade functions as a normal blade of its kind. In the hands of its intended wielder, the mind blade is a magic weapon that deals an extra 2d6 psychic damage to any target it hits.

I have emphasised "In the hands of any other creature, the mind blade functions as a normal blade" and "In the hands of its intended wielder, the mind blade is a magic weapon".

My question is: would the mind blade count as a magical weapon (or at least, does the slashing or piercing damage it does count as magical damage) for the purposes of overcoming damage resistances in the hands of someone other than its intended wielder?

Ordinarily I would assume yes, since a magic weapon is still a magic weapon even if you haven't attuned to it. But given that it specifically calls out the fact the it's a "normal blade" in the hands of someone it's not intended for, and that it's a "magic weapon" in the hands of its intended wielder, it makes me doubt my default assumption. Also that it's gone out of its way to describe something outside of the usual language of attunement (i.e. it could have said "while attuned" or similar like the descriptions of many other magic items do, but instead it defines an "intended wielder").

Obviously either way the not-intended wielder cannot attune to it or deal psychic damage with it; this is just about it being a magical weapon for the purposes of overcoming damage resistance.

No, because that would be to "function" as a magic weapon, which it explicitly lacks the ability to do

The key (as I see it) is parsing the word "function."

Generally, the function of something is its capacity to act or behave in a certain manner, or to have a certain purpose. So an item's ability to be used in combat so as to overcome any specific resistance is certainly a "function" in the ordinary sense of that word.

Thus whenever the sword "functions as a normal (not magic) blade" then it lacks the functions that are associated specifically with magic ones, including the function you are asking about.

No it does not

As you already emphasis in the question:

In the hands of any other creature, the mind blade functions as a normal blade of its kind.

In other words it is not magical or special in any way. This is further emphasized by the sentence coming after it:

In the hands of its intended wielder, the mind blade is a magic weapon