Only the Crown Court can appeal to the Court of Appeal
If the defence is appealing to the Court of Appeal against sentence or conviction it must be lodged verbally or in writing within 14 days of the trial and there must be a notice of appeal filed within 28 days of conviction and you must get leave
There is opportunity for cases to be reviewed later by the criminal cases review commission (there been more than 100miscarriages of justice that have been quashed by them in the last 3 years)
If the prosecution is appealing the the Court of Appeal then it must be on a point of law against and acquittal (witnesses were nobbled or new and compelling evidence or public interest) or against an unduly lenient sentence