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On the phone I use https://tasks.org/ to take new items or work with todos through the caldav interface (but that's android, guessing something similar exists for Apple stuff)
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organice
An implementation of Org mode without the dependency of Emacs - built for mobile and desktop browsers
For mobile I've been using organice, and synching through gitlab https://github.com/200ok-ch/organice
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Appwrite
Appwrite - The Open Source Firebase alternative introduces iOS support . Appwrite is an open source backend server that helps you build native iOS applications much faster with realtime APIs for authentication, databases, files storage, cloud functions and much more!
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Joplin
Joplin - an open source note taking and to-do application with synchronisation capabilities for Windows, macOS, Linux, Android and iOS.
I don't know how complex your Taskmamagement is. But if it's simple task management (Todo list with the option to add some text to each entry on a subpage) you're after and only the synching is a problem, you could have a look at "Joplin" https://joplinapp.org It also allows you to select an external editor on your pc, if you don't like the internal one, I think.