There isn't one answer, but 'schola' is the most common.
In specifically American contexts, it's specifically standard to use 'schola': Schola Latina Bostoniensis (Boston Latin School, 7–12); Schola Latina Puellis Bostoniensis (Girls' Latin School, 7–12); Schola Daltoniana (Dalton School, K–12); Schola Sanctæ Mariæ (St Mary's School, 5–12); Schola Præparatoria Georgiopolitana (Georgetown Visitation Preparatory School, 9–12); Schola Latina Roxburiensis (Roxbury Latin School, 7–12); Schola Sancti Pauli (St. Paul's School, 9–12); Schola Latina Brookliniensis (Brooklyn Latin School, 9–12); Covingtonensis Latina Schola (Covington Latin School, 7–12); Schola Amicorum Abingtonensium (Abington Friends School, 9–12); &c.
Because of the way the US uses 'college' to refer to the first four years of university, it's less common to use the more proper 'collegium' the way many British schools do. Still, you see it occasionally, apparently most often with Catholic schools: Collegium Aquinatis Sancti Bernardini (Aquinas HS, 9–12); Collegium Sancti Caroli Borromeo (St Charles Preparatory School, 9–12); Collegium Sancti Benedicti (St Benedict's Preparatory School, K–12).
Private institutions are also often partial to 'academia' (after Plato's old haunts): Academia Phillipiana (Andover, 9–12); Phillip. Exoniensis Academia (Phillips Exeter, 9–12); Academia Sancti Xaverii Cincinnatensis (St Xavier HS, 9–12).
Newer American schools who want to distinguish themselves from ludi litterarii or scholae grammaticales now sometimes use 'schola superior': Schola Superior Archiepiscopi Moeller (Moeller High School, 9–12); Schola Superior Universitaria (University HS, 9–12). You could follow the Latin Wiki and take that as a calque for 'high school' but it's worth noting that in Europe, the same term is more often used for colleges and universities.
Similarly, the Latin wiki will sometimes calque the modern terms as 'educatio' or 'schola secundaria', although in classical Latin the adjective would've carried a pejorative 'second-rate' association.
On the other hand, America isn't Europe.
'Lycæum' (after Aristotle's old haunts) is common in France but meant something else entirely in the US and isn't used in HS latinizations. The places where it has been used in English usually refer to vocational schools, not proper liberal-arts high schools; the places where has been used for actual high schools tend to be emphasizing their Frenchness.
Ditto the largely Mitteleuropan use of 'gymnasium', which would draw blank stares or jokes about PE if one attempted to use it to refer to an entire American school.