Weyl's Branching Rule for $SU(N)$-Setting

The question is answered on page 385 of the classical Zhelobenko book

Compact Lie groups and their representations

for the more general case of $SU(n+m)/SU(n) \times SU(m)$.

Maybe the following paper might prove helpful to your question:

Masatoshi Yamazaki, Branching Diagram for Special Unitary Group SU(n), J. Phys. Soc. Jpn. 21, pp. 1829-1832 (1966)

Every irrep of $SU(n)$ extends to irreps of $U(n)$, and conversely, the restriction of any irrep of $U(n)$ to $SU(n)$ remains irreducible. If your dominant weight of $SU(n)$ is $(a_1,\ldots,a_{n-1})$ then extend it to $(\sum_{i=1}^{n-1} a_i, \sum_{i=2}^{n-1} a_i, \ldots, a_{n-1}, 0)$, apply the $U(n)$ restriction, take differences $f_i-f_{i+1}$ of the resulting signatures.