The following test will verify that your NANO's I2C Logic is working alright (assuming UNO-Slave is healthy).
1. Connect your NANO and UNO as per following diagram (Fig-1).

Figure-1:
2. Upload the following sketch in NANO-Master.
#include<Wire.h>
void setup()
{
Serial.begin(9600);
Wire.begin();
Wire.beginTransmission(0x21); //roll calling for the Slave
byte busStatus = Wire.endTransmission();
if (busStatus != 0x00)
{
Serial.println("I2C Bus communication problem...!");
while (1); //wait for ever
}
Serial.println("Slave found!");
}
void loop()
{
}
3. Upload the following sketch in UNO-Slave.
#include<Wire.h>
void setup()
{
Serial.begin(9600);
Wire.begin(0x21);
}
void loop()
{
}
4. Check that the Serial Monitor of NANO-Master shows some diagnostic message.