Well I was chatting with the ol' man about this last month. He said I could put in the new tax disc before the old one expires (I am talking just days left here).
Anyway, I took out my new disc on New Year's Eve, and when I parked up, I stuck the new disc in there. If the Police pulled me over for displaying it too early, I would laugh at them and ask why they're not chasing the people displaying no tax disc!!
Anyway, I renewed my disc online last month. You have to have the new style MOT certificate to do it, but everyone will have one soon. Nice 'n easy.