Thank you, Tenno. We are aware of issues in The New War when playing with touch controls. We're working on these fixes, but in the meantime you can advance on iOS if you have a gamepad / controller (DualShock 4, Xbox Controller etc). If not, please wait for an upcoming hotfix!
If you are stuck inside the Quest, please get in touch with support at support.warframe.com for assistance! Thank you Tenno.