Is it legal to bundle the GCC cross-compiler (tar.bz2 file) along with a GPL v2 licensed ARMv4T processor core on Github? The compiler will extracted during simulations to compile C code for the core to execute.
Since processor core code does not appear to be in any way derivative of GCC, this sounds like a typical case of placing two independent programs side by side, which the GPL FAQ calls "mere aggregation":
What is the difference between an “aggregate” and other kinds of “modified versions”?
An “aggregate” consists of a number of separate programs, distributed together on the same CD-ROM or other media. The GPL permits you to create and distribute an aggregate, even when the licenses of the other software are nonfree or GPL-incompatible. The only condition is that you cannot release the aggregate under a license that prohibits users from exercising rights that each program's individual license would grant them.
The licenses of the two works do not interact if they are independent works. Since the license of each work independently allows you to distribute it, you may distribute both together.
If you were to combine GCC with the processor core in a way that somehow created a combined work under copyright law, then you could not distribute that combination: the GCC is under GPLv3, and versions 2 and 3 of the GNU GPL are incompatible.
If the processor core were licensed under "GPLv2 or any later version" then you could distribute the combination under GPLv3.