Going on vacation soon. What Camera should I buy?

The EM-5 blows the other two away.  Speed of focus is the first reason...the NEX-7 is not as good, the Fuji is horrendous.

Secondly, the 5-axis IBIS (on-board in-camera image stabilization) is an incredible advance in the video and still-taking world.  I shot video with it at full-telephoto (50mm with my Olympus 12-50mm kit lens), and the video looked like I shot it with a Steadicam (and I'm a retired Director of Photography, with multiple international awards, who's actually shot with a Steadicam).  Note that this was while shooting hand-held, with no object to brace against.

If you want a small camera, the E-M5 is the best combo combining a small camera with small lenses.  The NEX-7 will saddle you with poor lens choices, and gigantic boat anchors on the front of the camera.

I had the Nex-7 but recently replaced it with the OM-D. I find the OM-D to focus quicker overall. The touchscreen on the OM-D also allows for easy focusing - just like we can focus on particular objects on an iPhone by tapping it on the screen. 

I like taking pictures in manual mode, and find the mix of dials, buttons, and touchscreen capabilities to make more sense, where the Nex-7 lacks the touch interface and the buttons aren't organized as well - for example, you'll find yourself switching to record mode many times by accident. 

These two things are just announces I had with the Nex-7, first was how the SD card slides into the bottom, instead of the side - resulting in having to remove the camera from the tripod when needing to switch cards out (not a common thing with today's oversized cards), second were the filters. I prefer to take RAW images, the Nex-7 will not allow the use of filters when in RAW; the OM-D does.  

Then there's the lens thing - there aren't many e-mount lenses, but with the OM-D you can use Olympus, Panasonic, or Leica lenses (and others).