Simply extent this draw's function to draw something else once the initial slideshow is done:

1.  if(cinematic){
2.   image(images[counter], 0, 0);
3.    } else {
   
4.   // whatever the game would draw here.
5.   } 
6. }

I am not sure I fully understood your question. I don't see a class in your code. Well, not a custom one.
You have to keep this draw() method, and it must call the display / draw method in your class, usually to draw the associated object.

Oh, sorry for not being more descriptive, I want to use the class that i've pasted in another class. 

See if Why do I get a NullPointerException? article answers your case (ie. cine is probably not global or is shadowed).

I've read up on the NPE and it seems that whats causing the problem is 

Some progress! I have put the missing code from void setup, into the function I wanted to call which is the body.

Several issues the most important is that the loading of the images should be done in setup(). When a Processing sketch is started it executes the setup method once and then the draw method is executed approximately 60 times per second.

image(images[counter], 0, 0);
The problem with this line and NPE is almost certainly due to the fact that one or more of the image failed to load. If counter had been outside the array index range you would get an ArrayBoundsException instead.

In line 12 you are storing the current time in lastTime every frame so in line 14
millis() - lastTime will always be zero never >+ DISPLAY_TIME

I have modified the code you have provided below. It will stop displaying images after the last one has shown. I have added line 15 and if you remove the test (line 19) then you will get a looping slde show.

1. void setup() {
2.   for (int i = 0; i < images.length; i++) {
3.     images[i] = loadImage("cine" + nf(i+1, 2) + ".jpg");
4.   }
5.   lastTime = millis();
6.   counter = 0;
7. }
8. void draw() {
9.   background(255);
10.   fill(0);
11.   if (millis() - lastTime >= DISPLAY_TIME) { // Time to display next image
12.     counter++;
13.     if (counter == images.length) {
14.       cinematic = false;
15.       counter = 0;
    
16.     }
17.     lastTime = millis();
18.   }
19.   if (cinematic) {
20.     image(images[counter], 0, 0);
21.   }
22. }

Thank you for the help with clearing up my code however, this does not solve the issue of calling the draw function successfully from another class.

Again, unless there is a vocabulary misunderstanding, you don't call the draw() method of Processing (the global one) from a class. You call it from nowhere, it is Processing which calls it regularly, on each frame.
You rather create in the classes you make a method (call it display(), draw(), whatever) that you call from the main draw().