How to interpret this recipe for walnuts harvested at a given time?
I wouldn't get too hung up on such a specific date. The date given is likely because of it's proximity to Midsummer's Day celebrations. In Italy and France the harvesting of green walnuts was traditionally tied with St. John's Day.

The keys for harvesting green walnuts are:


Bright green exterior as walnuts will turn yellow-ish as they ripen.
Before the hard wooden shell develops, the nut should cut in half easily. I've seen some recipes suggest putting a pin thru the nut to test if it's still soft enough.
When the size is between a large olive and small egg


The only issue I could see with harvesting too early was that the size of the nut may be too small. Harvest too late and the wooden shell will start to form.

You'll probably have a window of about 6 weeks to find green walnuts that fit the conditions above. If you have your own tree you have the benefit of being able to see the nuts as they grow and mature. Otherwise you could try asking your produce provider when they usually have green walnuts to get a good idea of what dates are common in your area. I'm guessing to closely match the recipe you'll want to be on the early side of that 6 week period.