GrubeZebro

Really depends on the lenses you wanna use. The sigma 17-50 f2.8 for the nikon is really useful for example. You can adapt it to the Sony, but when I tried it last year, the camera just felt super off balance and hard to hold.

  

Sony a6000.  Focus magnification for manual lenses.  full-time EVF.  Can change aperture on fly.  
 
I might pick the D5300 is photography was my primary use.
 
I like the size of the a6000, but agree with Inazuma, it may be an issue for you so you should handle both cameras.

  

I would take the a6000, especially for a more video-centric use. But that's just me, and in your case, it doesn't matter what I'd take, or what anyone else would.
 
All these "cheap" cameras have some shortcomings, usually different ones in different models. They all are likely to show some moire and other digital artefacts at some point, so there's not much point in obsessing about it. In the case of the a6000 the most obvious shortcoming would be the lack of a mic input, apart from the one in the fancy hotshoe.
But whatever that shortcoming is, you just find ways to work around them, and try making the best out of your choice, whichever it may be in your case. You could agonise for months over the bits, knobs and specs of each camera, without getting any footage done. They all deliver decent footage in the right hands, but none of them deliver perfect footage. 
 
The nasty truth is, you may not even know what the real shortcoming of each given camera is in your case, until you have actually shot with one for more than just a few minutes. The spec sheets or even the online test videos are not likely to tell you that.
 
So my real answer would be just pick the one you fancy, the one that feels nice in your hand and the one you can afford, and then just go shooting with it. Your money, your decision. You have no obligation to justify your choice to anyone, apart from those with whom you live and share a household budget, perhaps. Seeking assurance from peer acceptance online is waste of your time.

Thank you all guys for answers!

I have had both cameras in my hands and I don't really care about the size.
When it comes to the system, my friends have Nikons, so we could share lenses. On the other hand I could use all my old m42 lenses with Sony via adapter.
I really don't like Sony's EVF but I know it might be useful.
I also don't mind not being able to change aperture in Nikon's live view.
In fact important difference for me is better image in d5300 (really love it) and peaking in alpha6000.

In general I would rather buy Nikon but I still have Andrew's unfavorable review http://www.eoshd.com/content/12111/nikon-d5300-review on mind.

Overall, having choice between Sony a6000 and Nikon d5200/5300, which one would you take?

Nikon offers better vid quality and codec but Sony has features like zebra and nice EVF.

There is probably mistake in the first line (not A600 but A6000).