Melksham is a town and civil parish on the River Avon in Wiltshire, England, about 4.5 miles northeast of Trowbridge and 6 miles south …Melksham is a town and civil parish on the River Avon in Wiltshire, England, about 4.5 miles northeast of Trowbridge and 6 miles south of Chippenham. The parish population was 18,113 at the 2021 census.