united-states
The General Rule
In U.S. law, most first direct appeals of trial court decisions which are "final orders" are a matter of right.
There is generally a direct appeal of right from a decision of a trial court which is not a court of record (generally a very low level limited jurisdiction trial court, such as a municipal traffic court or a justice of the peace court) by a trial de novo in a higher court of record, and there is also generally an appeal of right from the decision in the trial de novo made by the higher court of record.
Appeals from U.S. Bankruptcy Court decisions and from certain decisions of federal court magistrates, to U.S. District Courts (i.e. the main kind of federal trial court) follow the same pattern as appeals from trial courts not of record, with de novo review by the U.S. District Court followed by a direct appeal of right from the U.S. District Court ruling, even though U.S. Bankruptcy Courts and proceedings of federal magistrates in U.S. District Courts are both courts of record. Likewise, some, but not all, decisions of state court magistrates are treated for appellate purposes as decisions of court not of record.
There are exceptions to this rule, however, the most notable of which are set forth below.
Exceptions To The General Rule
Further appeals beyond a first direct appeal of a decision of a court of record, such as en banc review in a U.S. Court of Appeals (i.e. federal intermediate appellate court) of an initial thee judge appellate panel decision, or review by a state's highest court from an intermediate appellate court, is generally discretionary, and denial of appellate review at these levels is unreviewable.
Petitions for rehearing or reconsideration of a decision by the same court that made the original decision are generally discretionary.
A decision to allow an appeal filed after the deadline for doing so, that would otherwise be an appeal of right, is discretionary where it is allowed at all (often there is a jurisdictional deadline for filing an appeal that is a month or two after the usual deadline). This is also usually allowed only with a showing of "good cause" or "excusable neglect" for the late filing.
Most interlocutory appeals (i.e. appeals before a case is concluded) are allowed only with leave of the trial court, the denial of which is basically unreviewable, until a final order is entered in the case (at which point appellate review of the interlocutory order may be moot). A few select kinds of interlocutory appeals, however, such as appeals of the entry of a preliminary injunction, of denials of a claimed right to arbitrate a dispute, of contempt of court convictions, of 4th and 6th Amendment evidence suppression rulings in criminal cases, and of denials of a claim of immunity from being sued or prosecuted on a claim, are interlocutory appeals of right. The exact list of exceptions varies from state to state and between the state courts and the federal courts.
A U.S. District Court's (i.e. federal trial court's) decision to refuse to allow a case to be removed from state court to federal court, after this is attempted, is basically not subject to review in an appellate court. As explained here (explaining a U.S. Court of Appeals for the Fourth Circuit decision on the matter):
Under 28 U.S.C. 1447(d), “[a]n order remanding a case to the State
court from which it was removed is not reviewable on appeal or
otherwise, except that an order remanding a case to the State court
from which it was removed pursuant to section 1442 [federal-officer
removal] or 1443 [civil rights] of this title shall be reviewable by
appeal or otherwise.” The Fourth Circuit acknowledged that the
Supreme Court had interpreted similar language in 28 U.S.C. 1292(b) to
confer appellate jurisdiction over the entire order, rather than
particular reasons for the order, and that the Seventh Circuit had
relied on that authority in holding that 28 U.S.C. 1447(d) authorizes
appellate courts to review any issue in a remand order so long as
removal was premised in part on the federal-officer removal statute or
the civil-rights removal statute. But the court concluded that those
decisions were insufficient to abrogate preexisting Circuit authority
interpreting 28 U.S.C. 1447(d) as conferring appellate jurisdiction
over only the enumerated grounds for removal.
A U.S. federal court can ask that a U.S. state's highest court resolve a question of state law that is unresolved in the relevant state by posing what is called a "certified question" to the state's highest court. But the decision regarding whether a certified question will be presented by a U.S. federal court is discretionary and is basically unreviewable.
In the U.S. federal courts, appeals of certain kinds of prisoner's petitions, such a habeas corpus petitions, are generally allowed only with leave of the court, which is basically unreviewable. Some of the details of this are discussed in this Congressional Research Service report. This situation is most similar to the general rule which the question asserts applies in New Zealand.
I qualify some of these statements by saying "basically" because often a decision which is ordinarily not subject to appellate review can still be reviewed by the highest relevant state court, or in the federal system by either a U.S. Court of Appeals or by the U.S. Supreme Court, with a discretionary extraordinary writ such as a writ of mandamus (or a court rule providing for an equivalent kind of review such as Colorado Appellate Rule 21).
Also, while the rules set forth above reflect the overwhelming majority rule in U.S. states, there may be some exceptions in particular U.S. states to the general rule, particularly in states where there is no intermediate appellate court.
Utah's system is particularly unusual, with an appeal of right to the Utah Supreme Court in many kinds of cases, which in turn, either decides the case itself (a minority of the time) or refers the case to Utah's intermediate Court of Appeals (most of the time).
Historical Footnote
This was not always the case in appeals of federal criminal convictions in the United States. The direct appeal of right in criminal convictions entered in federal court dates only to 1891, when the intermediate U.S. Courts of Appeal were established.
U.S. states mostly started to allow direct appeals of right of criminal convictions in the same time frame, plus or minus a couple of a decades. Prior to 1891, federal criminal convictions could only be set aside through a writ of habeas corpus (on much narrower grounds than a direct appeal of a criminal conviction) or through a Presidential pardon (something that was granted much more freely as a result of the lack of a direct appeal remedy).
The right to a direct appeal of a trial court ruling is still not a constitutional right under the U.S. federal constitution in either state court or federal court cases, although certain constitutional rights (like the right of an indigent criminal defendant to have state appointed counsel and the preparation of a trial court transcript at state expense) do apply to direct appeals of right in cases where they are allowed by statute or court rule.
There is only a U.S. constitutional right to file a writ of habeas corpus challenging a criminal conviction or an incarceration not supported by a criminal conviction. Historically, this was largely limited to review of the jurisdiction of the court, the existence of a conviction, the legality of the law establishing the crime, speedy trial rights for someone who has not been convicted, and the duration of incarceration authorized by the sentence imposed upon conviction. These days, however, a habeas corpus petition can be granted on wider grounds than it could have been in the late 1700s and early 1800s.
Thus, a New Zealand style scheme for criminal appeals would not be unconstitutional is established by Congress or a U.S. state, under the U.S. federal constitution.
