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The short answer to this question is NO (to give a more precise answer one would have to go into much detail and invoke the classification of nilpotent orbits in simple Lie algebras). Probably, the quickest way to see this is to look at the last section of the Springer-Steinberg paper on conjugacy classes (published in Springer LNM, vol. 131). There one can find a description of the reductive parts of the centralisers $C_G(u)$ in the case where $G$ is a classical group. If we take for $G$ a symplectic group of rank $n$ (which is simply connected), then the conjugacy classes of unipotent elements in $G$ are parametrised by certain partitions of $2n$. The reductive part $L$ of $C_G(u)$ decomposes as a direct product of orthogonal ans symplectic groups. The derived subgroup of $L$ will not be simply connected if one of the orthogonal groups $O_m$ with $m\ge 5$ occurs as a direct factor of $L$. This condition can be described in purely combinatorial terms and if $n$ is sufficiently large there will be many such instances.