Conservation of linear momentum and conservation of angular momentum both follow from Noether's theorem, which shows how conservation laws arise from symmetries.
Conservation of linear momentum arises from translational invariance - that is, if you shift space in some direction, the laws of physics are the same. Conservation of angular momentum arises from rotational invariance - that is, if you rotate space in some direction, the laws of physics are the same.
Consider a particle moving in one-dimensional free space. The Lagrangian is
$$L(x,\dot{x},t)=\frac{1}{2}m\dot{x}^2$$
Taking the transformation
$$t'=t,\quad x'=x+\epsilon,\quad \dot{x}'=\dot{x}$$
This then gives
$$\delta x=\epsilon,\quad\delta\dot{x}=0$$
Noether's theorem states that there is a conserved quantity for some generalized coordinate $q$, which is
$$J=\frac{\partial L}{\partial \dot{q}}\delta q-F$$
That is,
$$\frac{\mathrm{d}J}{\mathrm{d}t}=0$$
Here, it is simple to see that $q=x$, $\dot{q}=\dot{x}$. Therefore,
$$J=m\dot{x}\epsilon-F$$
From here, it is easy to see that $m\dot{x}$ is a constant, that is, $p_x$ is conserved.
We can construct a similar Lagrangian for any number of bodies in any number of dimensions, and find that the total momentum of the system is conserved. Indeed, we can also do this for angular momentum. If we do so, we find that these two conservation laws are independent of one another. Any momenta is conserved, regardless of the other. Therefore, both are conserved, and neither can be "converted" to the other.