Conservation of momentum can be derived during collisions by using Newton's laws of motion. But in other cases, do we simply take it like an axiom ?
Answer
We don't take conservation of momentum as an assumption, and neither do we take Newtonian mechanics as an assumption. Instead the fundamental assumption we make is that the systems we study in classical mechanics obey the principle of least action.
It is hard to overstate just how important this principle is. From it we obtain Lagrangian mechanics and Newtonian mechanics, but it gives us a lot more. For example the relevance to your question is that for a system described by a Lagrangian Noether's theorem tells us that conservation laws are related to symmetries (of the action). If the system has translational symmetry, i.e. action is invariant under translations in space, then momentum must be conserved.
The principle of least action is a somewhat abstract approach for students beginning their study of physics, and you normally won't be introduced to it until you start university. However if you are looking for the fundamental assumptions involved in mechanics then this is it.
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