In the US, there's a proper way to fold the flag. From what I can tell, there is no equivalent method in the UK.
When did the US adopt this special method for folding the flag?
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This answer contains some conjecture. Perhaps somebody else will be able to provide better information.
The first piece of evidence comes from the Flag Institute (The UK’s National Flag Charity). It provides a guide to Flying Flags in the United Kingdom. On page 6, under the heading As a Pall for a Coffin it says
If the flag is to be retained by the next of kin it can be folded using the Royal Navy’s method shown here ...
The diagrams shown are very similar to the US procedure.
The next piece of evidence comes from researching the British ensign, the flag flown by the British Navy. There are actually three: the White Ensign, the Blue Ensign, and the Red Ensign. Wikipedia says
Prior to 1864, red, white, and blue were the colours of the three squadrons of the Royal Navy, which were created as a result of the reorganisation of the navy in 1652 by Admiral Robert Blake. Each squadron flew one of the three ensigns.
and goes on to say
early flags of the American Revolution were modified Red Ensigns. ... the Union Flag in the corner was replaced by the current stars in 1777.
So my conjecture is that the current method was adopted from the British Navy, along with the Red Ensign. So, to answer the question, identifying "the flag" to mean the Flag of the United States adopted in 1777, the date would be 1777.