Nintendo doesn't always go on the last day...
They always go on the first. They would go right before the floor opens to the public on a normal year.
Again, I don't think we can use previous E3s to predict the air date here. The conferences are the main and pretty much only attraction this year, not the showfloor demos, and booths afterwards.
So yeah, I could see them going first too, like
Noipoi
said.
They could also go on Monday.
I really doubt Tuesday, though: It gives them no room for a Tree House segment, interviews, demos, etc. They're advertised as big headliners of the event ATM, so I don't think it's super likely they'll go last.