If $u$ is uniformly distributed over the sphere, we can write it as $u=Uv$, where $U$ is a unitary transformation uniformly distributed over the unitary group. Then the quantity $|u\cdot v|^2$ is just the modulus square of a matrix element of $U$. So your question is: what is the distribution of the modulus square of a matrix element of a random unitary matrix?
Consider a column of the matrix as a random vector. The only constraint is must satisfy is that its norm must be unit, $\sum_{i=1}^N|z_i|^2=1$. In other words, the joint distribution of the entries is simply
$$\delta(1-\sum_{i=1}^N|z_i|^2)=\int ds e^{is(1-\sum_{i=1}^N|z_i|^2)}$$
The distribution of a single element, $z_1$ say, is obtained integrating over the remaining ones. Indeed, you get something proportional to $(1-|z_1|^2)^{N-1}$ in dimension $N$, which is your Beta distribution.
This is independent of $v$, so the distribution of $v$ is irrelevant.
A very similar argument applies to the real case and random orthogonal matrices.