I found a workaround by installing the Telegram app to my desktop computer and registering via that. The registration process is the same, and includes a SMS message being sent to the phone. This came through fine. Then, when I did the registration process from the phone, the registration code came through Telegram itself and appeared on the Desktop app, so I was able to complete the registration process on the phone.
Hope that helps anyone else who's been having the same problem.