There's a fork of Telegram on F-droid called "Telegram FOSS" that runs a background service for notifications. I've used it and the effect on battery drain is pretty minor, like maybe 2% per charge. Not sure if something similar exists for Signal.

I use MicroG on my phone, it's a basic FOSS replacement for GMS and makes push notifications work for all apps. It gets push notifs from Google's servers but you don't need an account. It doesn't use Google at all for location, which is nice as that's a highly invasive aspect of Play Services. Not all ROMs are compatible with it, I use it on Lineage

Telegram X works for me