Depends how severe your BED is. There is a spectrum of course and at one end, yes there are healthy babies. At the other, there will be hypertension, obesity, glucose spikes, etc, all of which could impact your pregnancy in varying degrees.
My sister still struggles with anorexia and bulimia. She will binge and purge then skip meals. I wasn't really around her for her pregnancy so I don't know how much of that occurred during the pregnancy. I know she struggled with the thought of gaining a lot of weight. Her son is physically healthy as far as I'm aware and no issues following birth. He is on the autism spectrum but I don't know if that can be attributed to her eating disorder or not, or was a factor.
Also, I don't think there is anything wrong with being autistic or on the spectrum, I am simply pointing out something that may or may not be related to/a cause for it developing in my nephew.
This was me in the beginning of my pregnancy as well I was nauseous and had headaches constantly and the only time I felt any relief from the nausea was if I was snacking on something. You could start by healthier snacks your appetite will start to settle for awhile it does get better