Hi yaman,
According to your description, the migrated tasks list didnt display in the source of the Timeline web part.
Per my knowledge, only the tasks lists which are created via using the built-in tasks app can display in the source of the Timeline web part. Except the tasks lists which are created by the built-in tasks app, for other tasks list(migrated tasks list or Workflow Tasks list), when we go to Modify View->Style, there is no Show timeline above the view.
More information about Timeline:
So, for this issue, I suggest you create a new tasks list using the built-in tasks app, then move the migrated tasks from the migrated tasks list into the new tasks using Content and Structure feature. Then use the new tasks list in the Timeline web part instead of the migrated tasks list.
Thanks,