I'm probably no more in the know than you. I read his garbage, compare it with court documents and the legal process and try to make some sort of sense of it.
For contempt you have an automatic right of appeal against the sentence and conviction. Whilst (unlike normal cases) you will automatically get permission, you still have to go through the formality of asking for it.
As far as I know, the only appeal that has been refused was his filing for Neelu because he filed it in the wrong court (on her behalf), evidenced by it getting a new case number.
He still hasn't run down the clock on his own. He has submitted it within the time period (which I believe is 21 days from sentencing).
Technically Neelu could still get her appeal as it's in the discretion of the court. However, the letter to her from the State's solicitor does seem to indicate that he, er sorry, Neelu was told exactly what she had to do and failed to do it so they may take a dim view of it.