Weatherhead joins from Bostik Premier strugglers Harlow Town, who he signed for in July 2018.
He is a young central midfielder, who has been on the books at Barnet, Ipswich Town, Tottenham Hotspur and MK Dons as a youngster.
He left the Hawks after finding his first-team opportunities limited to just five appearances.
Goalkeeper Aldridge joins from league rivals Tilbury.
He was formerly with Stanway Rovers, Braintree Town and Hullbridge Sports.
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