Fascinating stuff.
And, he's still His Royal Highness, but he can't tell anyone.
So, by convention, then, a not so royal person coming into the presence of a royal person should bow to the royal person. Will people still be expected to bow to him? And people ought to be bowing to Eugenie, but not to Louise or Zara?
I'm guessing not a whole lot of bowing goes on these days anyway, except probably to The Queen. Maybe to The Prince of Wales too, especially now that he's an old man.
The largest medieval recreation group in the United States is called the Society for Creative Anachronism. (It was a name that was pulled out of the air by science fiction writer Marion Zimmer Bradley, when it was a group of friends having a themed picnic in a park. They didn't know the name would still be in use 50 years later.) I've been a member since college days. There, we bow to "royalty", all the time, and people call me "Your Excellency", because I'm a baron instead of a plain old lord, but we know we're joking.
Back in college days, the group always had a "new member night" early in the semester, and people interested would introduce themselves and say a little bit about themselves. At one of those, a fellow stood up and said, in an authentic English accent, "My name is Warren Taylor, and I'm a Briton, and I'm here to find out why you people think this is anachronistic."