[RPG] Do Rogue abilities function in wild-shaped beast forms

The argument for "No"

Wildshape specifically bans spells (PHB 67):

You can’t cast spells, and your ability to speak or take any action that requires hands is limited to the capabilities of your beast form.

Thus, even if a spell theoretically doesn't have any verbal or somatic components, you can't cast it in your wild shape.

Are mystic disciplines spells? The UA says "kind of" (UA:Mystic 9):

Psionic disciplines are magical and function similarly to spells.

Based on that argument, one could say that you can't use psionic disciplines, because they are like spells and you can't cast spells while wildshaped.

The argument for "Yes"

Wildshape states (PHB 67),

Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. ...

You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.

Because psionic disciplines depend only on your mental abilities, and you retain your mental abilities while whileshaped, you should retain the benefit of your class feature.

We can compare this situation to a Warlock's eldritch invocations. While most of them are spells, some of them, like Gaze of Two Minds or One With Shadows, give you magical abilities that are not spells. Because you retain your non-spellcasting class features, you should be able to use those abilities while wildshaped.

Conclusion: Unclear, probably yes

I'd probably rule "yes" at my table, because I find the argument for yes more compelling. However, the evidence can arguably go both ways. It's important to remember that the mystic class is still playtest material. Jeremy Crawford, who makes official statements on rules, has stated that UA material isn't tuned for multiclassing at all. I expect that if/when the class is officially published, interactions like this will have been taken into account.

The druid retains any abilities that their animal form can use.

Wild Shape states:

You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. (PHB, 67)

Emphasis mine.

There's nothing that would prevent the druid from using Haste or Bardic Inspiration that has been given to them by someone else while in animal form, so they are able to maintain those effects and make use of them.