In the Specify Fields export dialog box, you will see a checkbox that says "Apply current layout's data formatting to exported data". You want to turn that ON.
Make sure that when you do the export, that you are on a layout that has your date field showing and it is displaying it in dd/mm/yyyy format. The date field will now export how you want it.
NOTE: It might make sense to create a special layout for exporting data from. All you really need on that layout would be your date field formatted the way you want it. If you have multiple date fields, just put all of them on this special layout and again, format them as desired.
Hope this helps.