Hello K Rees,
There is a well researched Family Tree for this line on Ancestry. They have many sources which include a baptism for Josia Ellies in Wynbunbury on 26/08/1631. Father: James Ellies. As you say Josiah went to London and there are many records for him there. One is of his burial in 1713 in Bunhill Fields. He was buried in the Quaker Burial Ground there. His Will is included in the sources on one tree.
Josiah married Mary Wilcox on 16/11/1697 in a Quaker ceremony. The marriage document is online and shows Josiah Ellis to be the son of James Ellis of Woodonbury in the County of Chester, schoolmaster dec'd.
He is shown to have had about 8 children. Their years of birth are 1662-1674, so he was married before possibly. The baptisms for the children are shown as having a mother by the name of Mary though.
In his Will Josiah states Mary to be alias Wilcox, alias Adams, daughter of William Adams late of Monmouth. It appears to have been her second marriage, probably his as well.
The Will is a very interesting read.
Name variants have been shown as Ellies and Ellice.
James Ellis is shown as born c1605 but no source for that.
There are sources for him being a Schoolmaster and residing in Wynbunbury in 1625,1631 and 1636.
James Ellis had other children whose baptisms are in the Cheshire parish records. They are: George Ellis, b1625 and Samuell Ellice b1636.
I would enquire of Chester archives as to what material they may have on this family.
Jo