If it is bullying to send an early hour email, that's a lot of people who bully then. They don't have to be opened.
The culture, and the bosses’ expectations is very relevant here. You can’t make any judgement about whether it’s reasonable or not without knowing that.
What we know about their office culture, from Harry:
“It didn’t help that everyone was working around the clock. There were so many demands from the press, such a constant stream of errors that needed clearing up, and we didn’t have nearly enough people or resources.
At best, we were able to address 10 percent of what was out there. Nerves were shattering, people were sniping. In such a climate there was no such thing as constructive criticism. All feedback was seen as an affront, an insult.
More than once a staff member slumped across their desk and wept.”
And from a former employee who spoke to Valentine Low, Times journalist:
When Meghan thought an employee had let her down, Meghan called them when they were out at dinner: ‘Every 10 minutes, I had to go outside and be screamed at by her and Harry. It was “I can’t believe you’ve done this. You’ve let me down. What were you thinking?” It went on for a couple of hours.’ The calls began the next morning and went on ‘for days’.
This is not a healthy culture being described here by either person. Emails at 5am in an unhealthy culture where the employers don’t recognise their employees’ boundaries can easily be seen as unacceptable.