Possibly the William Coxe, father of William Coxe in your question. Same county.
mentioned in the record of William Coxe and Elizabeth
||18 Aug 1595|
England Marriages, 1538–1973
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Citing this Record
"England Marriages, 1538–1973 ," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:NVRK-VQB : 10 December 2014), William Coxe and Elizabeth, 18 Aug 1595; citing Lambourn,Berkshire,England, reference , index based upon data collected by the Genealogical Society of Utah, Salt Lake City; FHL microfilm 251,838.