Telegram often won't work without a proxy — familiar. The fix is simple: add a proxy in the app settings. Go to Settings → Data and Storage → Proxy, enter address and port, done. Traffic goes through the proxy and the messenger works.
Which type to take? Plain SOCKS5 suits almost everyone. There is also MTProto — Telegram's own protocol, but it needs special servers. Practically, SOCKS5 is easier: it works with any app.
Speed matters for messengers: take a low-ping proxy, otherwise messages arrive late. 30-40 ms is enough for me. And avoid free public proxies — they are slow, unstable, and you don't know who runs them.