I woke up my daughter up at 4 am but she has a cold. Does this effect the EEG and will I get sent home after sleep depriving my child if she has a cold?

It should not. Having a cold should not affect getting the eeg. I cannot speak for the policies of the facility where you are having it done...I know a cold will cancel elective surgery but should not impact an eeg.
Should not be issue. In actuality, sleep deprivation is often employed as a technique to assure that paroxysmal activity is accentuated and diagnosed on the eeg tracing. Therefore, your situation should not in anyway prevent the procedure from being performed fully and properly.