I'm afraid you're chasing a chimera. The answer really is there in the Wikipedia article on the Channel Islands, although perhaps it requires a bit of unpacking. And some details are scattered among some other Wikipedia pages and other websites.
The UK Parliament website has the following:
The constitutional relationship of the Islands with the UK is through the Crown and is not enshrined in any formal constitutional document.
So if you are looking for a document, that document does not exist. It's like @MarkJohnson said (brilliantly IMO):
That is one of the secret ingredients of the Westminster system: nobody knows how it got there or where it is going.
The "British Constitution" is not a single document like the US Constitution. It doesn't even consist entirely of documents. Filling the gaps are "conventions", something I've termed "hidebound habits" in this other answer of mine. This is analogous to how "English Common Law" doesn't always have statutes backing it up.
Although The Crown has the ultimate responsibility for defending the Bailiwicks of Jersey and Guernsey as a feudal possession, this has devolved on the UK government in a vague, ill-defined way. The next sentences in that Parliament webpage:
The UK Government is responsible for the defence and international relations of the Islands, while the Crown, acting through the Privy Council, is ultimately responsible for ensuring their “good government”.
So, how did the situation get this way?
In the 10th century, a group of Vikings settled down in northern France. These Vikings, of course, became the Normans.
French Chronicler Dudo of Saint-Quentin says that in 911, there was a pact between French King Charles III and Rollo granting him the lands he and his companions occupied around the lower Seine. The conditions were that Rollo and his companions convert to Christianity, and also pledge to defend the lands they held. Rollo became the first Duke of Normandy.
(Side note: In 927, Wessex king Athelstan conquered Northumbria to unite England for the first time).
In 933, French king Raoul gave Rollo's successor William I Longsword the Cotentin Peninsula and the Channel Islands. William had just put down a revolt by some other Normans who thought him too Frenchified, and I suspect these were the lands the rebels had previously held.
The Islands became a feudal possession of the Crown in 1066 when "the Crown" became a possession of the Dukes of Normandy. In 1204, England lost all of the the Continental part of the Duchy of Normandy, leaving only the Channel Islands. England's possession of these islands was confirmed in the 1259 Treaty of Paris with France.
This being the Middle Ages, there were temporary occupations by the French and reconquests, and it became necessary to reconfirm the relationship between the different islands, and there were more than a dozen charters over the next 600 years, adding or revoking privileges for one island or another.
If you must have a breaking point, it will be 1689, right after the Glorious Revolution. At that time, Parliament was busy stripping away all sorts of prerogatives from the Crown. That year, there was an Order in Council that revoked a long-held privilege the Islands had held since a 1483 Papal bull: neutrality in times of war.
By taking away the islands' right to be neutral, the English (and by 1707, UK) Government implicitly assumed the responsibility for defending the islands.
The Order in Council may may even say this, but I can't find the text to it. It may have burned up in 1827. I do have this from a 1904 book on the Channel Islands' history, however:
The treaty of Neutrality had by this time become virtually a dead letter, and it was finally abrogated in 1689 by William III, on the ground that it formed an easy mean for James II to communicate with his partisans in England.
- The Channel Islands; by Carey, Edith F., A. & C. Black, London,, 1904, available at archive.org
(Carey's book refers to 1840's Casearea: The Island of Jersey for its information on Jersy's neutrality, which in turn refers to Gough's 1794 or 1806 edition of Camden's Britannia)
Alternatively, from this webpage, we can deduce that the islands' charters were simply not renewed, and only the privileges in the Bill of Rights applied to the islands, along with the rest the rest of the country.