The spells that come with feats Telepathic and Telekinetic can be cast with no verbal, somatic, or material component requirement. Would this mean that a wild shaped druid could cast these spells since all they would need is their mind to cast them?
You cannot cast spells while in Wild Shape.
The feature description of Wild Shape says:
You can’t cast spells [...]
So you cannot cast spells.
There is a way around this: cast them before using Wild Shape.
The rules for Wild Shape continue:
Transforming doesn’t break your concentration on a spell you’ve already cast, however, or prevent you from taking actions that are part of a spell, such as call lightning, that you’ve already cast.