From your research it would appear that Nicolas Johnson brought Eme and her son James with him to the Colonies in 1666 or shortly thereafter. (What happened to her son William?)
Peter was transported in 1661 by George Gouldsmith and earned his freedom and citizenship a few years later.It is interesting that the Fugates lived so near each other, but as you say, it may be just a coincidence as there is no evidence that he even knew Eme and James.
It could be that there are two different families that arrived in Maryland/Virginia with similar names that ultimately evolved into the Fugate that exists today. One descending from James ffugeete and Aveline Lambe (a French sounding name) and the other from Peter Fugatt(and a hundred other spellings), who almost certainly is French.
The descendants of Eme's son James appear to be well documented and the same can be said for the descendants of Josias.If only we could find the Peter-Josias connection. Good work Janice.