It's $U(5) \times U(1) \times U(1)$.

We have a natural map from the normalizer of $G$ to the outer automorphism group of $G$. The outer automorphism group of $G$ is $\mathbb Z/2$, generated by the inverse transpose / complex conjugation.

This map is trivial, because if it were nontrivial, an element mapping to $1 \in \mathbb Z/2$ would give an isomorphism between this representation and the complex conjugate  $\mathbf 5 \oplus \overline{\mathbf {10}} \oplus \mathbf 1$, which does not exist as these are not isomorphic.

The kernel of this natural map, for any $G$, is $G Z(G)$, where $Z(G)$ is the centralizer, since every element of the kernel acts by conjugation as an inner automorphism, i.e. is an element of $G$ times something that acts trivially by conjugation, i.e. is an element of $G$ times something in the center.

So the normalizer is $G Z(G)$. 

The centralizer is $U(1) \times U(1) \times U(1)$ since there are three non-isomorphic irreducible representations of $G$, so the centralizer of $G$ consists of matrices acting by scalars on these three representations. 

So the normalizer is the product, inside $U(16)$, of this $SU(5)$ and the $U(1) \times U(1) \times U(1)$ scalars. We can check that this product is isomorphic to $U(5) \times U(1) \times U(1)$ by letting $U(5)$ act $\overline{\mathbf 5}$ by the dual of the standard representation, $\mathbf{10}$ by the $\wedge^2$ of the standard representation, and $\mathbf 1$ trivially, and letting the two factors of $U(1)$ act by multiplication on the last two representations.