1) The chief current uses, that is orderly, neat, etc. are on record only from the opening years of the eighteenth century, and earlier meanings are: timely, in good condition, well-favoured, of good character.
Any of the latter might explain the by-name: 1379 Johannes Tydy, Grindleton. The word was also used as a cow’s name: 1486 to Agnes Chapman a cowe called Tydee, Felixkirk and this might share the same origin. Alternatively, a tidy cow in Scotland at that time was one that gave milk, and oxen could be described as ‘fat and tydye’.