It may not take months depending on how it started. I believe EC that is caused from lip biting will heal faster than if it is chemically induced (i.e. caused from accutane).
There is nothing stopping you from healing one lip at a time if it means you can have a life. I do understand the toll this takes and not everyone can live having both lips EC. You have to find a way forward such as this and not give up, because it doesn't have to be all or nothing.
Before you ask, there is nothing to say the upper lip won't heal independantly from the lower lip. If you can avoid applying anything and not worry to much about water and brushing your teeth/drinking/eating, this may actually be easier than worrying about these things, as managing them is a constant reminder everyday they have not healed.
I know from experience that gently exfoliating one lip while leaving the other alone looks a damm sight better than having EC on both.