Child protection is run by your municipality (sveitarfélag, pronounced roughly SVAY-tar-feh-lahg: the local council for the town or area you are registered in). Under Article 10 of the Act, every municipality is responsible for child protection and must operate a barnaverndarþjónusta (BAR-na-vern-dar-thyoh-nus-ta: literally "child protection service", the local office with the caseworkers). Small municipalities must join together so that each service covers at least 6,000 residents (Article 11). Which office handles your family depends on your registered legal domicile, your lögheimili (LURG-hay-mi-li: the address you are officially registered at), not on where you happen to sleep.
Decisions that go against a parent's wishes are not made by the caseworker alone. They are made by an umdæmisráð barnaverndar (OOM-dye-mis-rowth BAR-na-vern-dar: a district child protection board, a panel for a whole region rather than a single town, which decides the formal rulings) which, under Article 13, is independent of the municipality's ordinary administration. Above that sit Barna- og fjölskyldustofa (BAR-na og FYURL-skil-du-sto-va: the national agency for children and families) and the welfare quality and supervision agency, both listed in Article 3 as child protection authorities.
Appeals go to a national body, the úrskurðarnefnd velferðarmála (OOR-skur-thar-nemnd VEL-fer-thar-mow-la: the welfare appeals board, an independent committee that reviews decisions made about you). Article 6 states plainly that rulings and administrative decisions of the child protection service, the district boards and Barna- og fjölskyldustofa can be appealed to it. The heaviest decisions do not stop there: they go to a court.