This was largely state by state not a federal law and took place between 1852 (Massachusetts) and 1918 (Mississippi). My expertise is in the history of education in Virginia so that is the state I can talk about in detail here. To understand the compulsory education laws its important to go a bit into the background of public education in the state as its all connected.

As a requirement for ending Reconstruction, Southern states were required to set up a system of free public schools. So the Virginia Constitution of 1869 included a clause requiring the establishment of free public schools in every county by 1876. Specifically it said counties were required to have "a uniform system of public free schools, and for its gradual, equal, and full introduction into all the counties of the state." The system was set up as segregated, as an amendment that would have prohibited racial segregation by requiring schools to be free to any student no matter what “race, color, or any invidious distinction” did not pass (Virginia Convention 1867). ( source )

Public schools were not eagerly adapted though, especially by whites because they were viewed as something imposed by outsiders through Reconstruction and an overreach of the state. There were many newspaper articles at the time calling them a Yankee invention or imposition. One opponent in particular was against them because he believed it would result in compulsory education which would threaten parental authority (Link 1986).

But, over the next few decades white Virginians began to be in favor of public schools. This was reflected in the annual Superintendents Reports. For example, in 1879 all of the superintendents were asked "History of public sentiment concerning public schools in the county (or city) during the year" and responses were generally in favor, although varied widely. They included:

"Public sentiment has continued to be practically unanimous in favor of public schools.

There has been little change to note. A few of the opponents of the system, seeing no way to abolish it, have settled down to make it as good as possible.

Public schools have grown in favor with the masses of the people, white and colored, but there are a few bitter white opponents; some against the present system, they say, and others against all education at public expense.

Attendance wasn't great during the first few decades, especially in rural areas. Many students had to work and other times long walks to school made it difficult. Going back to the 1879 Superintendent's Report, Albemarle County (where Charlottesville is located) had only 32% of white and 25% of black students enrolled in public schools and average daily attendance was 19% and 16% respectively. Just under 20 years later in 1906 Albemarle had 43% and 46% of white and black students enrolled and 25% and 28% daily attendance.

Another force working on public schools in the first decades of the 20th century just before compulsory attendance was school consolidation. This meant closing small local schools that often times had a hard time maintaining the legal average number of students to stay open (22 students) and consolidating students into larger central schools that they were bused to every day (often free buses for white students while black students had to pay for buses). This was popular with educators and some parents, but unpopular with others who no longer had nearby schools they could walk to and was seen as another over reach of centralized state/county power over local power. Consolidation really took off after 1920 but began around 1900.

Finally getting to your question, compulsory education was introduced in Virginia in 1908, on the later side for the US but definitely not the latest. Virginia’s law had a local option meaning that counties had to vote on accepting the law before it was enacted in a locality (Report of the Federal Security Agency, Office of Education 1908). It wasn't until 1922 that a compulsory education law that actually had a real effect was put in place. I can't seem to find that law right now, but that is what my main source (Link 1986) keeps saying. If I find it I'll edit it in. A change to state mandated rather than locality controlled in the 1920s must have happened even if I can't locate it as during the 1950s this law was repealed returning compulsory education laws to local control as part of Massive Resistance. The 1922 law also required counties to take over administering education taking that power away from local school districts, something I've seen reflected in school board minutes I've looked through.

So td'lr: 1852-1918 across the US. VA in act was 1908 but in practice 1922, with a break in the 1950s and 60s due to Massive Resistance. This was all part of larger education movements from parental control to state control over the 19th and 20th centuries.