BuddyPress.org

After looking into this in more detail I discovered the problem is caused by a potential hosting compatibility issue with BuddyPress.

As soon as I turned off the BuddyPress plugin WordPress was able to send out emails again.

I installed BuddyPress on a WordPress install on a different server with a different hosting company and it worked fine. BuddyPress didn’t stop emails being sent out from that hosting account.

The compatibility issue is with BlueHost.com where BuddyPress prevents emails being sent out. They use cPanel whereas the other hosting company where BuddyPress works doesn’t.