If I pick a fixed $U(1)\;\;$ subgroup in $SU(N)\;\;\;$, say a circle in the diagonal, I get the following action of $U(1)\;\;$ on $SU(N)\;\;$:

$U \to g U g^{-1}\;\;\;\;\;$ where $U \in SU(N)\;\;\;\;$ and $\;\;g \in U(1)$

The corresponding coset $SU(N)/U(1)\;\;\;\;\;$ is pretty simple for $N=2\;\;\;$ it is $SU(2)/U(1) = CP^1\;\;\;\;\;\;\;$. Is there a similarly simple description for $N>2\;\;\;$? What is the topology of the coset $SU(N)/U(1)\;\;\;\;\;\;$ to begin with for $N>2\;\;\;$?

orbit space, not acoset space. Also, you are mistaken about the quotient space $\mathrm{SU}(2)/\mathrm{U}(1)$ in this case. It is not $\mathbb{CP}^1$; it's actually a $2$-disk, i.e., a manifold with boundary. (The boundary is the set of 'orbits' of the fixed points, i.e., the elements of $\mathrm{U}(1)$.) Similarly, the other orbit spaces you want to study will be 'singular' as well. $\endgroup$