In the PHB, it mentions that upon rolling a 20 on a death saving throw, you regain 1 HP. However, are you still unconscious after receiving this 1 HP? Additionally, if magically healed (by cure wounds or whatever), do you gain consciousness immediately?
Yes, if you regain hp either by rolling a 20 on a death save or by any other method, you will gain consciousness immediately.
If damage reduces you to 0 hit points and fails to kill you, you fall unconscious (see appendix A). This unconsciousness ends if you regain any hit points.