I tend to agree with PeterH. Even though, you are not attaching wires to the CNC, and instead using a servo to press a button or operate a joystick, make sure your boss is comfortable with your gadget before attaching that. To that end, it may make sense to make your project more finished looking, that your normal Arduino setup with wires running out. Otherwise it will look like a bomb or hacker tool like seen in too many bad movies. Depending on the equipment, there may be legal issues of putting anything on the machine.
Ok, assuming you can actually put your gizmo on the CNC machine, is it doable? From what I understand of what you want to do yes it is doable, but is it doable by somebody who is just starting out in a finite amount of time? Perhaps, perhaps not. If you've never done a big programming assignment before where you are creating a new program, rather than just taking an example program and modifying a simple variable, you will learn in this project, but it will take longer. After being a programmer for 34 years, I know how to go about creating the program, but honestly, I don't know how to teach somebody to think like a programmer. My experience is there are some people that grasp it quickly, and some take much, much longer, and often need somebody locally to guide them.
You have at least several steps, and I would recommend tackling each separately, rather than trying a big bang theory where everything must work at once.
The first step is to read a button press and rig up a servo to press a different button. Here, I would use a potentiometer or trim pot on an analog input pin, so that you can control the position to position the servo, rather than iterating, changing the value in the program, and downloading it. You want to use an actual button, connected to a self contained light setup (or a second arduino) rather than just blinking a light, since the point of the exercise is to actually press a button. It will make things simpler if you get a button similar to the button on the CNC machine. You likely will want to keep the variable control, so that you can adjust things when you get to the shop. Then if you need to press the button for a set period of time, do that here.
The second step is to operate the joystick. You probably need 2 different servos, one to operate in the x axis, and the other in the y axis. You really want to get a joystick like your cnc machine to use in your test setup. I tend to think a second arduino would be useful here, and that arduino reads the value of the joystick, and prints it out to the serial monitor. That way you can calibrate the movement of the two servos. Again, if you need to use timing controls, you would add it here.
The third step is doing the display. If you have a second Arduino, you could put the display on that Arduino for starters, to display the x/y axis, etc. You want to iterate on the display until you find something you are comfortable using. I suspect a graphic display is probably more useful than a text display, but the programming is harder. Note, I have discovered that servos on the Uno disable PWM on two pins (9 & 10), and some LCD shields want to use 9/10 to control the display. If you need to have the display, then you probably want to go with a display that is connected via i2c or spi, instead of using direct pins.
The fourth step is putting it together. One of the key things to think of is when you have multiple controls, you almost never want to use the delay function. Instead you want to study the blink without delay example, until you understand it. You will need to write the parts as a state machine, so that every step, you say is it time to do action X, Y, or Z?
If you are wanting to have several trimpots to control the degrees of freedom, you may find an Uno doesn't have enough pins. There are various ways to multiplex pins, etc, but it may be useful to think about going to a Mega which has more pins.
Be sure to back up your sketches, particularly when you come to a stopping point, it is useful to have backups. That way you can always go back to a know position.