I do think psychic powers would continue to work, as they are "triggered" when they are cast and then has a set duration. So, most psychic powers will either require a friendly or enemy target, which has to be met when the power is cast. After that moment, it doesn't matter whose side the unit is on - it gets the benefit or drawback until the duration ends. An example of this is that a psychic power will continue to work for the set duration even if the caster is killed after (or by) the power being cast.
As for <TRAITS>, such as <LEGION>, <GOD>, <REGIMENT>, <CHAPTER>, <DYNASTY>, etc those should continue to work, as they only require that you have a detachment all with the same keyword (or with another permitted keyword - Astra Millitarum works that way). When you take control of it, that doesn't change - or if it does, it does so in your favour (one could argue that when you take control of a unit, you are forming a new and temporary detachment with just it).
And for auras, it needs to meet all conditions of that aura. Given that most auras either only effect friends/enemies and/or require a given keyword, a mind controlled unit is unlikely to benefit from any auras.
It does look like one needs to look at the given effects on a case-by-case basis, to determine whether they continue or take effect.