A code commit from Microsoft mentions an update will be coming out in June 2021. Windows Latest speculates this is the date it will be rolled out to the general public and that the launch itself may happen in May. Alternatively, it's possible that this is not about 21H1 but about the 21H2 update:
If you have been following the release cycle of Windows updates, you might have noticed that the first major update of the year is released in the month of May. For example, Windows 10 version 1903 and version 2004 updates were released to consumers in the last week of May.
Therefore, it’s very likely that Windows version 21H1 will be released in May 2021 and it will be rolled out to more users in June 2021.
However, at the same time, it’s also possible that Microsoft is talking about Windows 10 21H2 “Sun Valley” update, which is expected to hit the RTM (release to manufacturer) status in June.