> I find telegram messages only get insta-delivered about half the time. The other half, they're delayed till I next re-open the app. Doesn't seem to happen with any other app.

That happens to me with WhatsApp. Curiously a message to Telegram kicks it back into gear too. My fiancee first messages me 'calling' on Telegram before calling on WhatsApp, because she figured out (remotely, by experience) that otherwise it often doesn't ring.