In physics, the expectation of a random variable under the bra-ket notation is $\langle u \rangle$, how do I write the conditional expectation of $u$ on another variable $v$? $\langle u | v\rangle$ seems like an inner-product, so has to be something else.
The question seems to confuse classical probability theory with quantum theory. Random variables are classical, but bra-ket notation is quantum. Either way, $\langle u\rangle$ is not braket notation. I am not sure whether I have ever seen $\langle u | v\rangle $ used for a conditional expectation of a classical random variable, but if it were used the context would make plain that this is not braket notation, and that it is not an inner product.
You don't talk of random variables in quantum mechanics. You have the expectation of an observable operator, $K$, given the normalised ket $|f \rangle$
$$\langle K\rangle \equiv \sum_i k_i P(k_i|f) = \sum_i \langle f|k_i\rangle k_i \langle k_i|f\rangle = \langle f|K|f\rangle$$
where $k_i$ are eigenvalues of the operator, and $|k_i\rangle$ are eigenkets. The RHS is braket notation. The LHS is not. You can read the RHS as the expectation of the result of measurement of $K$ given the results $|f\rangle$ of previous measurement (i.e. measurement of the initial condition).
The operator $K$ can be written in braket notation
$$K=\sum_i |k_i\rangle k_i \langle k_i|$$