This question on Quora says,
Although the title is now one of status and prestige without any
expectation that the appointee will actually act as counsel to Her
Majesty, the appointment still appears to be referred to as an
“office” (see Vote Bundle Text (990511-06)). If regarded as an
“office”, then the appointment of QC/KC would not be affected by the
demise of the Crown. This is because the Demise of the Crown Act 1901
dispenses with the need for the fresh appointment of any office upon
the demise of the Crown. So no new letters patent would have to be
issued.
and
Take for example Sir Robert McCall, pictured below, who took silk in
1891, under the reign of Queen Victoria – thus making him a QC.
Subsequently, when Victoria’s reign ended, he became a King’s Counsel,
adopting the postnominals KC. There seems to be no record of a new
letters patent being issued.
This question on Quora says,
Kings Counsel all immediately became QCs when Queen Elizabeth became
Queen.
and notes one exception
By default, when the gender of the monarch changes, EVERY past
reference to Queen becomes King at once, or King to Queen as
appropriate.
Her Majesty the Queen ordered that the King’s Troop Royal Horse
Artillery should continue to be called the King’s Troop in memory of
her father.
This article relates to the death of King George VI and is probably the best authentic account available if what actually happened:
In London’s High Court, King’s Counselor Harold Shepherd had just
finished cross-examining a defendant when the news came. The court
adjourned. Ten minutes later, the lawyer resumed the floor as Queen’s
Counselor. Painters at another London court set to work painting out
the sign “King’s Bench” and replacing it with “Queen’s Bench.”