Telegram Desktop is Telegram's cross-platform client, and unlike what the name implies, it mostly works well on phones.
Telegram Desktop is available from Alpine's community repository and can as such be installed directly via apk:
# apk add telegram-desktop
By default, Telegram Desktop may overflow and have some of its UI elements appear off screen. This can be worked around by settings a sub-100% UI scale. Unfortunately, Telegram Desktop's in-game settings do not allow for this, however the command-line switch -scale is offered for setting an arbitrary scale, for example:
$ telegram-desktop -scale 80
This will set the UI scale to 80%.
To make this permanent, you can copy the .desktop file into your home (this is to ensure that updates don't overwrite your changes):
$ cp /usr/share/applications/telegram-desktop.desktop ~/.local/share/applications/
Then open ~/.local/share/applications/ in your text editor of choice and edit the Exec line to include -scale 80 (or whatever scale works better for you). It should look something like this:
Exec=telegram-desktop -scale 80 -- %u
By default Telegram Desktop uses its own notification system where notifications appear as windows. This doesn't work great in Phosh (and presumably other phone interfaces) and will result in flickering and the currently viewed application being switched out when a notification arrives. For a better experience you can enable native notifications in Settings -> Notifications -> Use native notifications (right at the bottom of the page). This will make notifications appear as regular notifications in your user interface (if it supports them).