Can my connected wheel work without using my phone?

Yes!

As soon as you start pedaling, the engine starts (unless the last setting you pressed was "stop").

The wheel remembers the last assistance level you chose. For example, if you used 50% assistance level on your last ride, it will be automatically used by the wheel on your next ride.

This also applies if on your last ride you cut the electric aid, it will remain cut until you use your phone again to restart the electric assistance.