[cannibalise a PS4 and something on the wii side]How eventually is it possible?
Search around for controller mods on essentially any console and you will find ones to do turbo buttons, remapping buttons, macro inputs (though these are marginally more complicated), external buttons for disabled persons controllers/arcade type setups/fight sticks and such.
These will rely on one of two concepts.
1) Debug points/test points. Controllers need to be tested in the factory so nearby the buttons on a controller will tend to be small points that both can take a signal in (confirming the brains part is working for that button) and show signal out when the button is pressed.
2) Buttons are usually little more than glorified switches. That is to say they are pressed or the they are not. Slice into the traces and solder wires on and you can throw buttons wherever. This tends to be more used where there are no debug points (vanishingly rare as testing is kind of useful) or they are in too awkward a position to get wires to and still have a working controller). Scraping solder mask and soldering to traces is obviously more annoying than soldering to a nice fat test point that is there ready to take solder but if needs must.
On the flip side the nature of switches (and possibly debug points) also works the other way. Slice out the brains of the operation* leaving the whole board as little more than a glorified switch box. This is how some choose to adapt old console controllers to the PC
https://learn.adafruit.com/usb-snes-gamepad/disassemble-the-snes-controller
It is much the same whether you are doing a dpad and single button controller for some hypothetical old console or a 600 key keyboard.
*at this point we could cover the brains of controllers. Typically you have serial, parallel and more complicated network/packet type arrangements. Those little magic adapter boxes will need to know things here and adapt signals accordingly, the little teensy++ noted on the link above being able to do this. However this sort of thing you are dodging by cannibalising a wiimote, nunchuck, cheapo/broken classic controller or whatever.
The principle works however for this as well, and might not even need a little brain box unless you want one for turbo, easy remapping (you can do basic remapping easily enough with switches), macros or the like. The main trouble however is likely to come in analogue sticks. There are two broad schools of thought for analogue sticks, though you will also meet another for the wii as it is not strictly an analogue stick.
a) Some kind of resistance. If you have ever done the resistor divider voltage dropper in electronics classes and expanded that to include a variable resistor to change output then pretty much this. Usually carbon traces and some kind of wiper, how stick drift mostly happens and why opening up some controllers looks like someone ground a pencil lead up inside it.
b) Some kind position sensor. Can be a switch (several hundred individual lines to measure where something is), magnetic (series of magnets or possibly strength thereof), optical (see scroll wheels in your mouse or the little strip of what at first looks like clear plastic that runs the length of the average inkjet printer (actually is an optical encoding strip that the sensor can determine position in).
The wiimote is not an analogue stick and more of an absolute position sensor (why you can spin the thing around) based on gravity and some kind of motion sensor inside it, plus the whole infrared camera on the front tracking the position of the sensor bar (sensor bar is some infrared LEDs -- point a digital camera at one if you have never done it, also why putting a couple of candles in front of the TV works instead). That said if you are using a wiimote in you can figure out how to adapt that to fit in your PS4 controller shell or whatever you are going with.
For the sake of being complete then every stick will have one reading for X and Y directions (4 sensors for your basic 2 stick setup), also why an analogue stick might fail in only a given direction.
Anyway custom resistances can make faking different readings/adapting things more fun but nothing too challenging as you can measure it easily enough and dial in the ranges a bit.
Digital signals for position sensors get more fun. Here you will have to learn both the necessary inputs if it is a powered affair and the output formats for all aspects to convert between things (digital to digital but a different type, digital to analogue or whatever other combo you like).
Said digital thing would also be how you fake continuous motion -- presumably leaving the controller full tilt will have the signal faked to say it is constantly moving (think being put on the end of a drill). Might need some custom tweaks for this one.
Alternatively you can consider ripping out the sticks from the host device and putting them in place of the sticks for the PS4 controller in this case. 3d printing, filing, cutting and more for this one, choice of whether to leave the PS4 stick caps or wii offerings, with the potential trouble of the springyness of the stick not being the same (or needing work to match) as well which could well have been half the point of this project.
Do also look up what a dead zone is -- everything from basic human vibrations to wear to thermal effects mean when the stick is at rest it is probably outputting a signal still saying I am moving, to that end controllers (and those looking to fake them) will have a range when the stick is at rest in the top of the controller**.
**or if you have ever held a gamecube controller stick a given direction when powering on/plugging it in and noted it always goes that direction (or maybe the opposite as neutral on the stick is now the opposite direction) you have met the sensed variety.
Anyway as you are merely taking signals in and out, maybe some adaptation, this resulting contraption will work on anything that works with the baseline controller regardless of whether it is hacked, with no need for adapter boxes, or the in the case of remapping either the console is primed to remap or the game is (most old consoles don't have enough of an underlying operating system to do anything so game level it is, which is its own kettle of fish).
Short version. Buttons on controllers are switches. Easy enough to replace one switch with another.
