The BBC is entitled to appeal, just as is the case with anyone involved in a court case. However, having read the judgement I don't think an appeal will achieve anything. The evidence given by senior news staff suggested they had lost sight of the distinction between things that are in the public interest and things that are interesting to the public. The statement today by Ms Unsworth, the BBC's Director of News, suggests she still hasn't got it and the statements by some such as the Society of Editors shows that they similarly don't understand the existing law or, indeed, the findings in this case. Tim Shipman of the Sunday Times, for example, has tweeted that "the police chose to put the issue in the public domain" which is directly contrary to the judge's findings.
A number of people on this thread have referred to West Yorkshire Police. They were not involved. The investigation concerned was carried out by South Yorkshire Police.
South Yorkshire Police are the most culpable over this in my opinion, though the BBC not without blame
The judge disagreed with you which is why the BBC has to pay 65% of the total damages (i.e. both the amount awarded today and the £400k already paid by South Yorkshire Police). The actions of a reporter led the police to believe that they had no choice but to co-operate with the BBC or there was a risk that their investigation would be compromised by premature publicity.
Why isn't he in prison with Tommy Robinson
Because he didn't commit a crime. What happened was a breach of civil law, not criminal law.
apparently the BBC reporter and staff tried to mislead the court, and were roundly slapped down by the judge
It is clear from the judgement that the evidence given some of the BBC's witnesses and the case they tried to make was not consistent with emails and notes at the time of the events.