Dear Ivy. I'm very sorry hearing you're refused. My husband's naturalization's case was the same as yours. We were going to send his application through nationality checking service. Although they said as you had 2 driving offences, one of which ending with fine (30 GBP) your application would be refused, we sent it by ourselves and in page 13 of application clearly explained why he had been fined in the court. We sent it 12 months ago and we got their response 2 months ago which was the approval of his case (thanks God!).
Unfortunately you would not be able to appeal but you can ask them to explain why you had been refused. In their website it has had been mentioned they're not gonna accept any appeal. If you had been fined you need to wait for 5 years from the date of conviction.
Hope all of the guys who are gonna apply for naturalization be careful in their driving as driving offence is a major offence (rediculousely) and will cause refusing your application.
Good Luck!