Java 7, 207 206 171 139 bytes
void c()throws Exception{System.out.print("La La Land");Thread.sleep(1000);System.out.print("\rL̶a̶ ̶L̶a̶ ̶L̶a̶n̶d̶ Moonlight");}
I'm kinda cheating with this first answer, because I use strike-through unicode.
Explanation:
void c() throws Exception{ // Method (throws is necessary due to Thread.sleep)
System.out.print("La La Land"); // Show initial text
Thread.sleep(1000); // Wait 1 second
System.out.print("\r // Move 'cursor' to the start of the line so we can overwrite the current text
L̶a̶ ̶L̶a̶ ̶L̶a̶n̶d̶ Moonlight"); // and print new text
} // End of method
Java 7 (with AWT), 444 429 341 bytes
Crossed out 444 is still regular 444 ;(
import java.awt.*;import java.text.*;void m(){new Frame(){public void paint(Graphics g){g.drawString("La La Land",9,50);try{Thread.sleep(1000);}catch(Exception e){}AttributedString s=new AttributedString("La La Land Moonlight");s.addAttribute(java.awt.font.TextAttribute.STRIKETHROUGH,1>0,0,10);g.drawString(s.getIterator(),9,50);}}.show();}
Since Java console doesn't have any markup like strike-through, you'll have to use Java AWT. And well, if you thought Java Console was already verbose, then this is even worse (I know, I know, most of you couldn't even imagined Java 7 being any worse..)
Explanation:
import java.awt.*; // import used for Frame and Graphics
import java.text.*; // Import used for all AttributedStrings
void m(){ // method
new Frame(){ // Frame
public void paint(Graphics g){ // Overridden paint method
g.drawString("La La Land", 9, 50); // Show the initial text
try{
Thread.sleep(1000); // Wait 1 second
}catch(Exception e){} // Thread.sleep requires a try-catch..
AttributedString s
= new AttributedString("La La Land Moonlight"); // Object to add markup to text
s.addAttribute(
java.awt.font.TextAttribute.STRIKETHROUGH, // Strike-through attribute
1>0,//true // Mandatory parameter before we can specify the length
0, 10); // From length 0 to 10 (length of "La La Land")
g.drawString(s.getIterator(), 9, 50); // Show this new text with strike-through part
} // End of paint method
}.show(); // Show Frame
} // End of method
Output gif:

La La La(nd)
pattern \$\endgroup\$