My 6 year old daughter Caroline had an EEG earlier this month to look for abnormal brain activity as part of a whole series of tests because of concerns withher development. (Turns out the EEG showed she has benign epilepsy - she has never had any seizures, to my knowledge.)
This is in the US, but the neurologist gave us instructions to bring her in with no more than 4 hours sleep, putting her to bed at midnight and waking her at 4am. (The test was at 8am.) She did much better than I could have hoped for. I'm a single parent, but I have a wonderful supportive boyfriend who stayed with us for the week. I palmed my other daughter off on her aunt, and my boyfriend and I put all our concentration into Caroline.
We took her out for a late dinner, followed by ice cream, the night before, then I stayed up with her to watch her favourite movies, while bf got some sleep. It wasn't until about 11:50 that she got really sleepy. My bf woke her up at 4, while I slept for another hour. He gave her a shower (the hair needs to be clean for an EEG) and then the 2 of them went for a walk in the dark with torches. (She thought that was great.)
The doctor's office is a 40 minute drive away, so I sat in the back of the car with her and had to poke her a couple of times to keep her awake. We had breakfast at a cafe near the doc's office, and then it was time for the appointment.
She was so tired that she lay still and willingly let the technician put the electrodes on her scalp, and she fell asleep during the test as she was supposed to.
I found that, by preparing for and anticipating the worst, it went a lot smoother than I had hoped.
Good luck.