What ended up being the problem, if you don't mind my asking?
Well, our mail server could not send out because it was blacklisted for "spam or rate of sends". H&E had been operating just fine for months after the update without any problems. I contacted support, and they promised to look into the issue and update me.
However, they were quiet for over a week, so I decided to take matters into my own hands and found a solution in Mandrill. It was highly recommended by other IPB community owners, and it's perfectly suited to handle the outgoing e-mail of busy websites. Their servers are reputable, and the set-up works very well. Their stat-tracking is also way better, as it allows me to see exactly how busy H&E's mail gets and shows any delivery errors (such as e-mail addresses that are no longer valid).
It may cost me a little extra money, but Mandrill's fees are very reasonable. Most importantly, the e-mail works!
I will be looking into a server move as well, since we are in danger of outgrowing our current hosting package. I have a holiday coming up in a few weeks, so hopefully I can find a good host and also launch a new Fanfiction Library then.
Yes, it works now!! Thank you very much
Glad to hear it.