Alternatively there's many bridges these days so you can use Matrix with one of the many apps supported.
IRCCloud does. Or if you want an open source stack: Soju as bouncer, Gamja as web interface, and Goguma as Android app. If you have a paid Sourcehut account you get Soju+Gamja hosted for you at https://chat.sr.ht/ .
I'm not sure The Lounge and Quassel support push notifications, but they otherwise fit your requirements.
You replied to my post where I said I switched to IRCCloud and the person asked for self hosted alternatives to it :P