Had the same issue during the beta but reckoned I'd just wait tile the end of the month. Checked on multiple android devices at home, reset the devices, reinstalled app, nothing changed, I can login through a google account alright, but no through jagex account, tapping it does nothing except a flash of 'trying to connect' then back to the main screen.
I have the same problem since the new client update. I tried everything; reinstall app, delete cache, rebooting phone etc. Nothing worked unfortunately, i think it's some technical bug in the app. Google log in works fine. I have a Samsung Galaxy S9+ 6GB RAM and android 10. So that isn't the problem. I hope Jagex fix this problem soon.