It is the rule in NY that a plaintiff may sue its attorney for failure to appeal or perfect an appeal. The burden of proof is no different than for any other case: did the attorney deviate from good practice, was that deviation the proximate cause of damage, and was plaintiff damaged?
The difference is that in NY, it is always a bench trial. Questions of whether there would have been a different result on appeal is never a jury question, and always a decision for the judge. Today Illiois Legal Malpractice blog reports similar decisions nationwide. Details.