If I can clarify Daves point re lenses with aperture.
They can both use lenses with apertures rings.. but those lenses are probably completely out of the scope or realm for your nan!
We're talking multi thousand dollar lenses.
The only limitation re the lenses is that they HAVE to be AF-S types. 
That's the internal mechanics of the lens focusing system .. Nikon call it AF-S. 
Some other's call it hypersonic(HSM) or ultrasonic(USM) or whatever focusing system, but it's an all electronic method for focusing. 
The reason I explain this is that within the nikon camera is a focus MODE(not system) called AF-S too.  
AF-S focus mode(AF-S = single shot focus mode) is where the camera focuses only the once. By default the camera will give a short beep when it's focused. 
Another focus mode is AF-C(continous focus mode) .. this allows the camera to continually focus until the shot is then actually taken. 
So an AF-S lens is totally different to these focus modes in camera. 
Nothing wrong with those lenses, and they are the default now for Nikon's consumer priced lenses. 
But if ever buying lenses, you need to be sure that it is an AF-S type lens, otherwise it won't auto focus with a D3xxx or D5xxx body. 
In terms of size and weight, they are for all intents and purposes much of a muchness!... actually overall the D5600 is smaller(but we're talking one gram less, or 1 mm less here) .. overall consider them the same.
The bigger difference tho in terms of size is the thickness of the body(where it matters). 
The D5600 is thinner by a small margin, but where this body thickness is, give the D5xxx body a massive advantage to its ergonomics. 
The body thickness difference is right near the grip, where your fingers get wrapped around, and back into the body(between lens and grip) .. on your right hand. 
Far more comfy and holdable than the D3xxx body.
I struggle to hold a D5xxx body myself(too small), and I can't hold a D3xxx at all. 
I was once asked by a couple to take a photo of them with their D3xxxwhatever .. and I had to hold it like a point and shoot. 
I got my daughter a D5500, and I just find it uncomfortable. 
But in saying all that, I'd foget the D3400 and D5600 altogether. I my mind a bit of a waste of money compared to a D5500, or if budget is a factor, a D3300. 
Unless you can't live with Nikon's idiotic Wifi(Snapbridge) power wasting feature. 
That's really about the only update that they each got over their respective older versions.
D5500 and D3300 can still be had brand new, even tho the newer models have been around for a while now. Sensors are the same as their predecessors too .. in fact sensors are both the same for both body types. 
You can get better lenses than the three you've listed, but when I got the D5500 for daughter, I thought for an extra hundred or so, better to get her a lens she can easily hold rather than use some of my heavier lenses all the time.
A few years ago I got my son an 180-105VR for use on a D70s and D300(my old cameras) as it was cheap and very easy to fix 'WHEN' he broke it.. turns out he was a good kid, and didn't brake 
But my experience with the 18-105VR was pretty good, so I took a punt and got daughter the 18-140VR kit lens. 
Even better than the 18-105!
So happy with that. 
One thing I can recommend for a non photography type is to get a lens that minimises their need to change lenses! .. so I'd highly recommend the 18-140 over the twin kit lens deal. 
I think it's actually more expensive for the single lens deal, but it's the better lens overall. 
If the need ever arises that nan ever wants to photograph small tweety birds or whatever and wants more reach .. then a better alternative to a xx-200mm lens and those not so good 18-300 lenses exist.
I'd recommend the AF-P Dx Nikkor 70-300 f/4.5-6.3 ED VR lens.
Note that there are a couple of models of this lens type that are similar. Don't try to save money with the non VR model, and the other one is a larger $1K model .. so I'm guessing not something of interest. 
The point of those two lenses(18-140 + 70-300 mm lenses) is that they then have a lot of overlap in terms of focal length. This minimises the need to change lenses due to the "I can't get close enough" syndrome. 
In the comparison link that Bytemrk linked too, there is a bit of an error in specs.
The list the D3400 as 395g and the D5600 as 465g. Those values are as per spec, but not equalised properly. D3400 spec is without battery and D5600 spec is with battery. 
Add the battery(which is the more important spec to be mindful of and the D3400 becomes 445g, so only 20g lighter than the D5600. 
There are a lot of points for and against on that comparative site and a lot if it is meaningless. ISO dynamic range bit depth .. will be the same unless your nan uses a specific software to edit the raw files!
The key important point differences are the articulating screen on the D5xxx bodies, and (if ever needed in the future) the 14bit bit depth for the raw files. 
It has a larger buffer(internal memory) so if nan ever shoots a teeny tweety bird and shoots 5 frame per second, she may get 2 or 3 seconds of images, rather than just 1 seconds worth on the D3xxx bodies. 
So, overall favour the D5xxx body over the D3xxx type. 
And if nan(or anyone else) doesn't really need immediate access to facebook or instagram or whatever social site to upload images as they are shot(ie. the wifi connection thingy) .. then favour the D5500 over the D5600. 
Noting that wifi is a massive drain on the battery too!
Note that D5500 also has wifi, but can be 'properly' turned off, so can be set to not draw power at all. Wifi is fairly useless tho, old tech and painfully slow and tedious to connect.
				
			 
			
		 
			
				
			
			
				Arthur.
All these discos are giving me a heart attack!
'99 D1 300Tdi Auto ( now sold :( )
'03 D2 Td5 Auto
'03 D2a Td5 Auto
			
			
		 
	
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