Career saved games :

Heya @suttyjosie, sorry for my late response,

This is possible from the Xbox one console...

Push Xbox button so your in the main menu, highlight need for speed tile, press  start (the one with the three lines) , it will bring up a menu, go down to manage games, this will bring you into need for speed options. On the right will be tiles for each of you Xbox accounts saved data, highlight the one you want, press start, click delete save data and choose delete from everywhere. Your game can then be restarted!

More info is here - http://forums.xbox.com/xbox_forums/xbox_support/xbox_one_support/f/4270/t/1637887.aspx

Thanks to Stacinator8977 for the solution.