What happens if the other party is not served by the court date

I am not sure about Nebraska, but in Wisconsin, a defendant in a lawsuit is not obligated to appear prior to being properly served with service of process. A plaintiff cannot obtain a judgment against you in Wisconsin at least, before the defendant is properly brought into the suit via service of process.

I was able to locate the following laws concerning service in your state:

25-505.01. Service of summons; methods.

Unless otherwise limited by statute or by the court, a plaintiff may elect to have service made by any of the following methods:
Personal service which shall be made by leaving the summons with the individual to be served;
Residence service which shall be made by leaving the summons at the usual place of residence of the individual to be served, with some person of suitable age and discretion residing therein; or
Certified mail service which shall be made by
within ten days of issuance, sending the summons to the defendant by certified mail with a return receipt requested showing to whom and where delivered and the date of delivery, and
filing with the court proof of service with the signed receipt attached.
Failure to make service by the method elected by the plaintiff does not affect the validity of the service. Source: Laws 1983, LB 447, § 22; Laws 1984, LB 845, § 21. Cross Reference: Workers’ compensation cases, manner of service, see sections 48-174, 48-175, 48-175.01, and 48-190.
Case Note: Plaintiff may elect to have service made by any of the methods specified in the statute. West Town Homeowners Assn. v. Schneider, 221 Neb. 674, 380 N.W.2d 265 (1986).

25-507.01. Summons; proof of service; return date.

Within twenty days after the date of issue, the person serving the summons, other than by certified mail, shall make proof of service to the court stating the time, place, including the address if applicable, name of the person with whom the summons was left, and method of service, or return the unserved summons to the court with a statement of the reason for the failure to serve.
When service is by certified mail, the plaintiff or plaintiff’s attorney shall file proof of service within ten days after return of the signed receipt.
Failure to make proof of service or delay in doing so does not affect the validity of the service. Source: Laws 1983, LB 447, § 24. Cross Reference: Workers’ compensation cases, manner and time of service, see sections 48-174, 48-175, 48-175.01, and 48-190.

Note, that if you voluntarily appear, it seems that under your state laws this contitutes service:

-516.01. Voluntary appearance; special appearance.

The voluntary appearance of the party is equivalent to service.
Prior to filing any other pleading or motion, a special appearance may be made for the purpose of objecting to the jurisdiction of the court over the person of the defendant. The defendant’s assertion of a claim for affirmative relief by way of counterclaim, cross-claim, or third-party claim waives any objection that the court erred in overruling the special appearance. The defendant’s participation in proceedings on any issue other than jurisdiction over the person waives any objection that the court erred in overruling the special appearance except the objection that the defendant is not amenable to process issued by a court of this state. Source: Laws 1983, LB 447, § 32.