I am basing this on a piece of information from the gaf hardware thread that was locked months ago, there was a user there who had said that Nintendo had made a deal with an American chip maker to produce their CPU/GPU for their next hardware. This would be around the end of last year, which lines up with the timeline.
I am also basing this on the assumption that Nintendo's handheld and console will be scaled hardware in the same family, that Wii U backwards compatibility is going to continue to be a goal and that Nintendo will release their new hardware in late 2016 (handheld) and 2017 for console.
I think Nintendo's partner is pretty obviously AMD, they offer GPU compatibility with previous consoles, they have been great with pricing and Nintendo has a long history with them. I think Wii U's CPU might come along as a co processor for handling OS and BC, but will not be used for future software. I think what they could do that would be even cooler is to push all of the Wii U architecture into the gamepad and run it reverse to the TV, but could be costly / impossible.
I look at AMD's APU line and their 2016 K12 announcements and see handheld gaming at it's best, Nintendo might even decide to go with ARM K12s here as they will offer everything Nintendo needs.
What I think we will see from the specs is fairly basic:
Handheld 2016 is 20nm for AMD...
CPU: 2 or 4 "cheetah" ARM cores 800mhz to 1.2GHz
GPU: 2 CUs @ 500 Mhz 128 GFLOPs (GCN2)
RAM: 2GB
Screen: 480P to 540P resolution
Price: $169-199
A single chip solution will drive price far down, especially overtime. 3DS sales stalled but the price of the hardware is still high, especially outside of 2DS, this I argue is a large reason why 3DS will simply not break 100m, and can be avoided in future hardware by allowing the price to line up with DS. This also would thanks to in part resolution and in part, improved GPU, give a very close/better performance over Wii U, in a handheld that should be able to sip less energy than the current 3DS.
Console 2017 (still likely 20nm though 16nm and 14nm is possible from AMD at this point)
CPU: 8 Cheetah cores @ 2+ GHz
GPU: 16 CUs @ 1+ GHz (GCN2) (2TFLOPs or more)
RAM: 8GB DDR4
Storage: 64GB /w external storage available
Price: $199 to 299
A single chip solution again will drive the price down, a co processor espresso would only cost a few dollars a sku and would offer benefits for BC, especially to their VC. However since they will likely use a gamepad still, I imagine they could potentially shrink the Wii U down enough to fit inside the controller for this next console all together. This would be ideal as while it would add cost to the unit, it would insure BC and give Wii U life going forward. However this might be impossible though a cool idea imo.
These 2 devices would run the same software at different settings, Nintendo would also only have 1 dev kit running at different speeds depending on the mode and allow developers to cash in on both platform's customers with a single product. Nintendo themselves would create some titles that would work on both platforms and some that are sister titles, like smash for 3DS/U but with the same code, with graphical settings changed. They would also have some games that are purely exclusive to one device. (think OoT/Majora's Mask, or Galaxy 1/2)
They should also try to create studios/buy them, in the west and allow NoA to manage those teams, while also expanding their 2nd parties/3rd party partners with first party IP creations.
I don't think anything I said is really far fetched, it requires will that Nintendo might not have, but I still would like to see this future and I don't really see a problem with the console specs for the main core, but it is completely adjustable, meaning you could put as many CUs as you want in there, I originally thought 20 was more likely, but I think the minimum here is 16, it is still a large increase over XB1 and far superior to the "increase" Wii U brought over last gen consoles. This is a speculation thread though, so feel free to come up with your own.
I avoided talking too much about where Nintendo currently is, they have a lot of problems with their Wii U, but the biggest is that Nintendo's software trickles out because they simply can't push 2 platforms at once, this is the one place where they have already committed to solve in the future with a single architecture and Iwata is on record saying that if customers only buy one, they will move to a single platform in the future (gen after next I imagine) so while I think Nintendo's Wii U won't recover nearly enough to make it a profit, I also think that they are learning from this failure and planning their next line better.
They could simply upscale the Wii U and release a handheld that is basically a Wii U in the future, and this would be a fast track to a more stable future, but I really believe AMD APUs in the long run offer more to Nintendo and the gamepad housing a Wii U, is a perfect solution to their BC needs.