The name is derived from "Oswald" and "Twistle". The word "twistle" is an old English word meaning "brooks meet". Legend has it that Saint Oswald, King of Northumbria passed through, giving the area its full title of Oswald's Twistle, which in time came to be Oswaldtwistle.
However, it is more likely derived from the name of the Anglo-Saxon who farmed the land. There is more information on the Wikipedia Page.