Depending on your mobile provider you could probably email-to-text. It's not very popular nowadays so most people have forgotten it exists but most providers have some kind of email address format that lets you send an email and it'll come up as a text on your phone.
For instance, AT&T's is [email protected] and T-Mobile's is [email protected].
So if you had something like GMail you could setup a filter to auto-forward all GitHub notifications to one of these email-to-text gateways your provider has setup. It's not a push notification but it would serve the same purpose of notifying you each time you got a notification from GitHub.