In answer to the question posed in the title: very small. I went to my carrier’s store today to get my account switched over to a Nano SIM ready for my iPhone 5′s arrival tomorrow. Once I got it home I decided to poke it out of its supporting card to compare it with the Micro SIM.
Leaked images in the past had made it seem not that big a deal, almost like size had barely changed. I found it to be a completely different story.
For reference: a 20p coin and the universal pinky were compared with the Nano SIM (placed on top of a Micro SIM). Just in case it was immediately clear: the pink one is the Nano SIM, white is Micro.
Again, leaked comparisons in the past have made it seem that the contact points on the cards were exactly the same size. Admittedly, this could be carrier dependent, but mine were massively different in size. Full gallery below: