The Mee family web site.
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Henry Le Mey
? ?
? ?
Thomas Mee
Elizabeth Mee
Robert Mee

There is a record of a Thomas Mee who died in 1557 but not before he had a son Thomas in 1552. Assuming he was 30ish when he started having children this would indicate he was born around 1520 and so it is unlikely he was John's brother. He isn't mentioned in John's will so isn't likely to be his son. Thus he must be John's nephew.