In the 1752 British switch to the Gregorian calendar, the day after 
Wednesday September 2 was Thursday September 14. Special days and 
celebrations in the ecclesiastical and civil calendars kept the 
nominal dates that persisted before the calendar change. So 
Christmas, Guy Fawkes Day, anniversaries (including of baptisms and 
births) remained on the same nominal date. The exception was made, 
however, to any dates affecting property rights so that landlords and 
employers did not lose 11 days. Settlement days thus preserved the 
traditional duration of the contracts and the "real", as opposed to 
the "nominal" date, by moving 11 days forward. So from 1752 onward, 
"New" Michaelmas day was celebrated in the church calendar on 
September 29 (the same nominal date as before 1752), but Bank of 
England shareholders, or agricultural workers who had been hired on 
Lady Day, were paid their dividends, or harvest wages, on the "Old" 
Michaelmas settlement day of October 11. Labor fairs also made 
preserved the "real/old" date; before 1752, the Lady Day fairs were 
on March 25th, but after 1752 on April 6.

So the anniversary of 5 June should remain so. A source of confusion, 
however, could arise for British anniversaries of events that 
happened between January 1 and March 25. Before 1752, the New Year 
began on Lady Day, March 25 and after 1752 on January 1. Some 
political economists such as John Graunt who wrote in the late 1600s, 
even used fractions such as in 1 February 1662/3 to reconcile British 
and continental differences before 1752.

Judy Klein