I really wish you could go to court. If you don't show up, they will probably decline your appeal because you didn't show up. I'm not sure if you'd need a solicitor if they are deciding on an appeal. It might be simply a case of approving your appeal, or declining your appeal.
Maybe just go to the court, go through the metal detectors, and wait for your case to be called. Maybe you could ask your partner to come with you for support. Sometimes, one has to do what one doesn't want to do in life.
How much is the fine? Is this a criminal court or a traffic court? Maybe you could try calling the court, trying to maintain your anonymous status as long as possible. If it gets to the point where you might be able to reschedule your court date, I would then tell them your name, and reschedule. I'm not sure how it works in the UK--but in the US, it's not easy to reschedule.
If this were a criminal court (at least in the US), a bench warrant for your arrest might be issued. You probably wouldn't be arrested right away--but the warrant would be out there.
If this were a traffic court (in the US, again), a traffic warrant would be issued. Theoretically, they could also suspend your license for non-payment of the fine.
I hope someone will chime in who has experience in UK courts. And in appeals such as yours.
Truthfully, the criminal justice system in both countries don't give a rat's butt about mental illness. I think you know that, since you were in prison.
If you could tell me the exact name of the court, maybe I could do some research for you.