Where can I find a proof that a $C^*$-algebra is nuclear iff it has completely positive approximation property?
An excellent reference with (relatively) detailed proofs is:
Nathanial P. Brown and Narutaka Ozawa. $C^∗$ -algebras and finite-dimensional approximations. Vol. 88. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008, pp. xvi+509.
In particular, have a look at theorem 3.8.7.