Issue to get updated values using Serial communication

What do you think this does?

Height_Value is a float, you then define a new variable with the same name but as a String, and try to cast the value of float Height_Value to String Height_Value, which won't work.

I used this version of your code (slowed down a bit, added Serial.print() to see what is happening):

//Height Sensor
float Height_Value;

void setup()
{
    Serial.begin( 9600 );
    Serial1.begin( 9600 );
}

void loop()
{
    // Height value 
    int sensorValue = analogRead(A0);
    Height_Value= sensorValue*0.04811;
    String Height_Value=String(Height_Value);
    Serial.println(Height_Value);
    Serial1.println(Height_Value);
    delay(1000);            
}

And got this output:

19:39:18.694 -> 
19:39:19.694 -> 
19:39:20.694 -> 
19:39:21.694 -> 
19:39:22.695 -> 
19:39:23.735 -> 
19:39:24.735 -> 
19:39:25.735 -> 
19:39:26.735 -> 
19:39:27.735 ->

So nothing basically.

I connected a pot across 0V and 5V, wiper to A0, and turned it up slowly, modified the code to get rid of the cast to String:

//Height Sensor
float Height_Value;

void setup()
{
    Serial.begin( 9600 );
    Serial1.begin( 9600 );
}

void loop()
{
    // Height value 
    int sensorValue = analogRead(A0);
    Height_Value= sensorValue*0.04811;
    //String Height_Value=String(Height_Value);
    Serial.println(Height_Value);
    Serial1.println(Height_Value);
    delay(1000);            
}
19:40:41.065 -> 0.00
19:40:42.065 -> 5.20
19:40:43.065 -> 8.27
19:40:44.065 -> 11.55
19:40:45.065 -> 14.05
19:40:46.105 -> 17.42
19:40:47.105 -> 23.14
19:40:48.105 -> 23.48
19:40:49.105 -> 23.43
19:40:50.105 -> 23.48
19:40:51.105 -> 23.24
19:40:52.105 -> 23.43

Note that I used Serial.print() to send the data to the serial monitor to see what is happening, you need to learn to do this, it's a basic tool to find problems. You should litter your code on both Ardunos with Serial.print()s so you can follow the data through from analogue port to the SD card and see what is there at every point.

As you are sending the data 10 times per second but only checking it once every 7 seconds that will fill up the buffer on the receiving Arduino. I don't know what the effect of this will be, but I can't imagine it to be good. I suggest you read: Serial Input Basics - updated