At a guess this appears to be a civil matter and it sounds like family law ! If the latter, I know little about it !!
Generally, serving a notice of appeal has the practical effect of staying a Circuit Court order(s) unless there is any specific provision that a particular order or element of an order is not stayed pending the hearing of an appeal.
An appellant cannot delay forever. Sometimes, delay is used as a deliberate ploy. The way to deal with that is to set deadlines for the appellant. You or your solicitor can demand of the appellant - through his solicitor - that the relevant documents are lodged within a certain time frame e.g. within three weeks of the date of this letter. This demand should be accompanied by a warning that if the matter is not dealt with promptly you will apply to the High Court to have the appeal struck out for want of prosecution and then do that if there is no response. BTW if the appellant is a litigant in person the principles are the same but they might get more leniency in terms of time.
Courts may grant reasonable extensions of time to comply with such requests or orders but a time will come when continued delay will not be tolerated and a matter may well be struck out for failure to prosecute. Paradoxically, this type of situation is one where the other party must be seen not to be acquiescing in the appellant's tardiness.
BTW if the other party is "slippery" make sure that the warning letter is sent by registered post !.