Immigrated to James Island, VA 1623 - no name of Ship.
On Census of James Island 1623/24

Adventures of Purse and Person: Virginia 1607-1625:
A Muster of the Inhabitants on the Neck of Land near James Citty taken February 4, 1624 John Reddish came in the (no name of ship):  corne 2 barrels;  fish 100;  powder 1/2 pound;  peece 1; sword 1.  Mr. Alnutt and his servant here planted reconed before the Muster of James Cittie.

Ancient Planters and Jamestown Homes:
An area numbered 22 on a map of Jamestown in 1624 shows the land was first owned by Lieutenant Batters, granted to him by Sir Samuell Argall.  Lieutenant Batters sold it to David Ellis, who in turn assigned it to John Radishe.  John Radish sold it to Sir George Yeardley soon after 1624.

The Complete Book of Emigrants, 1607-1660
4 Feb 1625, John Reddish is living in 'Neck of Land near James City'.

Cavaliers & Pioneers: Abstracts of the Virginia Land Patents and Grants, 1623-1800:
20 May 1637:  John Radish and John Bradwell had 16 acres in the Land of James City, part of land abutting Eastward upon the Land formerly in possession of Mary Holland;  thence &c. rung close to Goose Hill Marsh.  Due by deed of sale to the said Radish from John Baldwin, late of James Isl., gent., and 1/2 has been sold by Radish to Bradwell.

29 October 1642: John Resbury, 150 acres lying in Wards Cr., near the Mill, NNE on Cross Cr., dividing same from land of John Freeman, ESE on the Cr. over against land of John Reddish and SSW on Edward Clooynes land.  Records regarding the transportation of three persons.

The 1642 date above is the latest date I have for placing John Reddish in Virginia.  Would love to hear from other James Island researchers, and find exactly when he arrived and on what ship!