FWIW, a lot of cellular providers have an email gateway for delivering SMS messages. There's also paid SMS gateways, and options for providing arbitrary push notifications to smartphones.
This was a few (3) years past, but they accepted root@localhost sendmail messages just fine in most cases, and delivered alerts within a minute or two of sending. We didn't rely on this long term, but it was a "good enough" first pass.
I'd probably recommend using one of the gateways (or a more fully-featured service like Pagerduty) for more serious businesses, but for personal use (or where an outage detected the next day isn't crippling), it's remarkably useful.