The order itself
Section 711.002 of the Texas Health and Safety Code names, in order: a person designated in a written instrument signed by the decedent; the surviving spouse; the decedent’s adult children; the decedent’s parents; the decedent’s adult siblings; and then further degrees of kinship. The first person on that list who can be located and is willing to act holds the right to control disposition.
This is a legal right, not a courtesy. A provider who cremates on the authority of someone lower down the list while a higher-priority person objects is exposed, which is why they will not do it.
Why the first line matters most
The very first entry — a person designated in a written instrument — is the one families overlook, and it is the one that solves the most problems. Naming your own agent in writing displaces the whole statutory order. It is the single most effective step available to someone who expects disagreement, or whose closest relative by law is not the person they would actually choose.
The appointing an agent post covers how that is done.
When people at the same level disagree
Where several people share a level — three adult children, for instance — the practical rule is that a provider needs agreement, or at minimum no known objection. A single sibling who objects in writing will generally stop a cremation, because cremation cannot be undone and no provider will absorb that risk.
When agreement genuinely cannot be reached, the question goes to a court. That is slow and expensive, and it is precisely what a written designation prevents.
When nobody can be found
If the priority person cannot be located after reasonable effort, the right passes down the list. What counts as reasonable effort is a judgement call the provider documents carefully. This is one of the more common causes of delay, and it is worth telling the provider early if a spouse is estranged or a child is out of contact.