This determination of component groups goes back to Elashvili, but has been improved somewhat in work of George McNinch and Eric Sommers here. 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 centralizers correspond nicely in good characteristic..