Faktisk implies that the listener is assumed to be unaware of the information they are being told, and that this information might be a little surprising.
Jeg har faktisk eksamen i økonomi. I have a degree in economics, actually. Jeg kommer faktisk fra Australia. I'm actually from Australia.
Egentlig means originally, initially:
Jeg har egentlig eksamen i økonomi. Well, my main degree is in fact in economics. Egentlig kommer jeg fra Australia. I come from Australia originally.
Nemlig emphasizes that that sentence is an explanation of something that has happened earlier:
Jeg skal søke på den jobben - jeg har nemlig eksamen i økonomi. I am going to apply for that job, seeing as I have a degree in economics. Det er dyrt for meg å reise hjem. Jeg kommer nemlig fra Australia. It's expensive for me to go home, since I come from Australia.
From Exploring Norwegian Grammar by Kirsti Mac Donald and Marianne MacDonald
