Messaging apps are the reason many people turn on a VPN in the first place, but they behave differently from a browser. They keep a connection open all day, wake up for notifications and use different traffic for calls. That explains most "it works, but…" problems.
Why messages arrive late
Android aggressively limits apps in the background to save battery. If the VPN app is put to sleep, the tunnel drops and messages wait until you open the phone. Exclude both the VPN app and the messenger from battery optimisation (Settings → Apps → [app] → Battery → Unrestricted) and allow background data.
Use Always-on VPN
Settings → Network → VPN → the gear next to your VPN → Always-on VPN tells Android to restart the tunnel automatically, for example after switching from Wi-Fi to mobile data. It is one of the most useful settings for messaging.
Why chats work but calls do not
Text messages travel over ordinary encrypted connections. Voice and video calls usually use UDP, a different kind of traffic that some VPN setups or networks handle poorly. If chats work and calls fail, the tunnel is fine for messages; try another network before assuming the app is broken.
About built-in proxies
Telegram and WhatsApp both support proxies. A proxy from an unknown channel can see when and from where you connect, and some are run to collect data. Only use a proxy from a source you trust, and prefer one VPN for everything over a different proxy in every app.
"Connecting…" forever in Telegram
Force-stop Telegram after the VPN is connected so it opens fresh connections, and check that no proxy is still enabled in Telegram's settings from an earlier attempt. A stale proxy is a common cause.
RealGram
RealGram combines a built-in Freedom connection and its own messaging, so there is no second app to keep alive. Availability still depends on your network. Install guide.