The order is final 30 days after entry, unless it is appealed or a motion for new trial is filed. Once it is final you can get a writ of possession immediately. Defendants do lots of things to drag cases out. Sometimes it works.
This is not legal advice. You should always discuss the specifics of your issue in person with an attorney. Be aware that there are time limits on all claims that depend on the kind of claim, so do not delay in seeking an attorney.
Under the facts briefly presented - technically it sounds like the appeal was dismissed b/c the tenant/defendant failed to perfect the appeal. So the JP Judgment became final and the file will/should be returned to the JP to issue the Writ to be served by the constable.
Ask the clerk of the court (JP and/or County Clerk) when they will issue a Writ of Possession. They should be able to let you know the procedure and costs.
BTW- If the Judgment is large enough and/or you think the tenant/defendant may have non-exempt assets now or in the future, you may consider requesting an Abstract of Judgment and filing that in the County Deed Records.