Why not?

Because in most of those mails it was mission critical that I receive the response. The process grinds to a halt if I don't receive response, irregardless of anyone and anything else. Both Alex and I were tied to the same process and action. The priority of it is unchanged.

This doesn't mean that recipients can't feel additional pressure to respond due to the presence of another party.

"Irregardless" is a double negative (and not a proper English word).

Yes and no to that. http://en.wikipedia.org/wiki/Irregardless