The next handheld should have a top screen in 16:9 at either 540p or 480p. It will not have an x86 or x86-64 processor. The best guess is that it will have a ARM Cortex-A53 or A57. It will probably be dual core. It will probably be based on the
AMD K12 or one of it's cousins. It will be paired with AMD GPU cores. How many will be based on power draw. Unless batteries have a massive revolution in the next year, that will be the limiting factor.
The next Console will be based on the same CPU and GPU cores - it will just have more of them running at higher speeds.
As far as backwards compatability goes, I suspect that the focus will be on handheld titles. In fact, if the 9th generation Nintendo handheld is backwards compatible with DS, 3DS, and New 3DS titles, I suspect that the 9th generation console will be as well.
I don't know what to expect for backwards compatibility with the Wii U, unless emulating PowerPC in ARM is trivial. I suspect that Nintendo may have painted themselves into a corner on that, and would have to include a Wii U in the 9th generation Nintendo console.
This can be solved in a manner of ways. One of the easiest would be to port the Wii U libraries to ARM and get an almost free greatest hits out of it through recompilation and tweaking. They could even do an upgrade program. Send in your discs for a downloadable copy compatible with your new system for $5 a game.
Wii and GameCube should be something that can be emulated and fall into Virtual Console territory.
I think a key strategy for Nintendo could be to have running compatibility in the future. Let TVs top out at 4k (anything higher is kind of ridiculous outside of a proper movie theater). Let the systems that they put out in 2030 play cartridges from the DS in 2005, and everything in between. If they can run like that, then they should be able to produce newer systems every year or two and abolish the generational cycles.
The hardest part for Nintendo might be NIH syndrome, and this has nothing to do with hardware. I think that they need to be able to bring in technology from the outside. Can they bring in a hypervisor from somewhere else like Xen or SierraVisor? I think they need to be able to not reinvent the wheel. Can they ape online features that Xbox and Playstation gamers take for grated like proper voice chat? Can they bite the bullet and do system wide achievements of some sort?
Whatever hardware they put out, it will play Nintendo games. The bigger issue is about whether their internal politics will let them buy the pieces that they need where they've failed to innovate and/or fallen behind.