You cannot appeal either the conviction or the sentence. Appeals against magistrates’ courts decisions have to be lodged with the Crown Court within 21 days of conviction.
The magistrates will almost certainly have made a “collection order” when imposing the compensation and costs orders. This means that in the event of your failure to pay, action can be taken to recover the sums involved without the matter being brought back to court.
This usually means, in the first instance, that bailiffs will be asked to attempt to recover the sum. If they cannot (because you refuse to pay them and they cannot seize goods which you own which they may sell to cover the sum) the Fines Office will attempt to set up an “attachment to earnings” order or a “deductions from benefit” order (depending on your circumstances) so that the money is compulsorily deducted from your income.
If this fails the matter will be returned to court. You will be ordered to appear before an Enforcement court. The magistrates will enquire into your circumstances and the attempts made to recover the sum. They will determine whether you do not have the ability to pay or are wilfully refusing to pay. Assuming they determine the latter, and they believe that all efforts have been tried and failed ultimately you may be committed to prison. There is a scale of prison sentences which are imposed according to the amount owed. For amounts not exceeding £200 this is seven days. For amounts between £201 and £500 it is fourteen days.
Please note this is a brief summary of the processes involved. There are other minor steps which may or may not be taken depending on the circumstances.