Big-O Notation for Menezes-Vanstone Elliptic Curve Cryptography?
Your bounds seem to be correct, assuming $n$ is the length of $p$, though writing $T(C1) = O(\log n)^2$ is wrong on two counts. First, it should be $O(\log^2 n)$ or $O((\log n)^2$) on the right. While $O(\log n)^2$ has the same meaning, it's not a "closed form", rather like writing $O(\sum_{k=1}^n k)$ instead of $O(n^2)$. Second, $T(C1)$ looks like $C1$ is some parameter like the input length or the number of vertices, whereas you intended $T(C1)$ to be the time needed to compute $C1$. Better notation would be $T_{C1}$, say, and you could include the input length $n$ as a parameter.

Moreover, note that there are faster integer multiplication algorithms, and some of them are used in practice in cryptographic contexts (not necessarily the ones which are asymptotically best). So you can get even better bounds on the running time depending on the implementation.

The reason you couldn't find the time complexity is that the algorithms use a constant number of two very common operations, namely multiplication and inverse modulo $p$, so the running times can be read immediately by experienced practitioners. This is even the case with more complicated operations such as modular exponentiation.