Natural spell does not allow casting while in any other forms other than the Druid's wild shape. For example, it does not permit a Druid to cast while in Shapechange forms. As of v1.23, the Quickcast menu (if selected from the interface) bypasses this restriction, and also allows to cast spells in wild shape without Natural Spell.
The feat does not allow a Druid to cast Epic spells in wild shape, since these function as feats and are thus blocked from access while polymorphed. In analog, it does not allow a Warlock to use her Eldritch blast ability directly. However, as with spells from any other classes, all Invocations are available (Eldritch blast shapes and/or essences can thus be used).
Spells that affect Animals usually do not affect a druid in wild shape form, and Natural Spell does not change this. However as of v1.22, the spells Magic Fang and Greater Magic Fang form an exception for the Animal, Magical Beast, and Dragon shapes, as does Reduce Animal for the Animal shapes.