In section 1 of the following, I wrote down a proof of the result in question:
McNinch, George "On the centralizer of the sum of commuting nilpotent elements." J. Pure Appl. Algebra 206 (2006), no. 1-2, 123–140. [arXiv version]
The argument I gave is a less powerful application of some of the tools used in the nice answer given by BCnrd -- it is really just an application of Chevalley's Theorem (I used the form found in Springer's "Linear Algebraic Groups" which is good enough for varieties but not for application to schemes in general). And (again in contrast to BCnrd's answer) my argument used the fact that orbits are locally closed.
I am glad to have read what is probably the "right" level of generality for this argument found in BCnrd's answer.
If $k \subset K$ is an extension of algebraically closed fields, Prop. 4 of loc. cit. show that each $G_{/K}$ orbit has a point rational over $k$ (when $G= G_{/k}$ has finitely many orbits). This gives the bijection between orbits over $k$ and over $K$ -- which of course is already a consequence of BCnrd's answer -- as in Scanlon's answer.