Hello.
No, they don't use any wrappers in latest versions of H2, they all are handled by own implementations in H2.
Actually legacy java.util.* and java.sql.* data types have different bugs in Java; don't use them if you can.
Documentation of H2 also recommends JSR-310 data types:
java.time.LocalDateTime is also supported and recommended on Java 8 and later versions.