I'm not sure how long Melbourne in Yorkshire has been around, but it's probably a long time. The Melbourne in Derbyshire is arguably where Melbourne in Australia got it's name from since it appears to have been named after William Lamb, 2nd Viscount Melbourne, and Melbourne in Derbyshire is the family seat of the Lamb's.
Since Melbourne in Derbyshire was mentioned in the Doomsday book, it has a significantly earlier claim on the name than Melbourne in Australia (found around 1835). ie close to 750 years earlier.