This determination of component groups goes back to Elashvili, but has been improved somewhat in work of George McNinch and Eric Sommers <a href="http://front.math.ucdavis.edu/0204.5275">here</a>.   Your set-up is equivalent to studying the same problem for a semisimple algebraic group and its Lie algebra in arbitrary chaeracteristic, but good characteristic (including 0) is essential for getting uniform results.

In particular, the situation for nilpotent elements of the Lie algebra and unipotent elements of the group is essentially the same, by Springer's equivariant isomorphism between the two settings   The classes/orbits and centralizers correspond nicely in good characteristic.

P.S. Concerning structural information on the centralizers, you can also consult Roger Carter's 1985 book on characters of finite groups of Lie type.   There he includes a lot of details about the classes and centralizers in your question over an algebraically closed field.   Since there are only finitely many unipotent classes or nilpotent orbits (the same number in good characteristic), his tables provide a clear overview.    Fine points of structure are also treated extensively in the newer AMS book by Martin Liebeck and Gary Seitz.