How did the United Kingdom government come to be responsible for defence of the Channel Islands?

It was a response to developments in international law

The reason, or at least part of the reason, seems to be developments in International Law following the treaties known as the Peace of Westphalia (1648).

Neither England nor the Channel Islands were parties to the treaties but the treaties seem to have been the starting point for the development in International Law of the modern idea of states and the result that, in International Law, the United Kingdom is a state whereas the Channel Islands are not states. In the Barclay Brothers (No 2) case the UK Supreme Court commented:

Nevertheless, the Bailiwicks are not independent states in international law. The United Kingdom Government is responsible for their international relations and for their defence. But it is the practice to consult the Island authorities before entering into any international agreement which would apply to them. The UK has also undertaken not to act internationally on behalf of a Crown Dependency without prior consultation; recognises that their interests may differ from those of the UK (especially in relation to the European Union, of which the Islands are not members) and so it may have to represent them both; and supports the principle of the Dependencies’ further developing their own international identities...

If the Channel Islands came to be regarded as falling on the non-state side of the state/non-state division as that developed in international law following the Peace of Westphalia, it then would follow that only the United Kingdon could act formally for them on the international stage in, for example, signing international treaties, or being represented at the United Nations. If that was the practical reality on the international stage (whatever the UK or Channel Islands may think about it) the UK is compelled to represent the Channel Islands in formal foreign relations and that creates, by virtue of necessity, an internal constitutional convention that the UK is responsible for the formal international affairs of the Channel Islands.

From that it must have followed that the UK government was also responsible for the defence of the Channel Islands since the conduct of international relations always has defence in the background - the extent to which a state is willing to take positions in international relations which might give rise to a military threat from other states depends, among on things, on its defence capabilities and its willingness to deploy them.

So the constitutional convention that the UK is responsible for the defence of the Channel Islands would appear to have emerged gradually, in the century and a half following the Peace of Westphalia, as an internal constitutional response (after the transition to constitutional government in Great Britain commenced in the 18th Century) to the emerging international law recognition of states.

Comment on the theory that the constitutional principle dates from the time of William of Orange

Historically the Channel Islands benefitted from the privilege of being able to continue to trade in time of war. This privilege ceased in 1689 when William of Orange became the King of England, and England, as a Dutch ally, went to war against the French. It has been suggested that this was the origin of the constitutional principle that the UK government (or English government as it would then have been) has responsibility for the defence of the Channel Islands. I am not convinced by this theory because at this stage in history the King would have defended his possessions simply because they were his possessions - a constitutional principle could only have meaning after the transition to constitutional government started in the 18th Century under the Hanoverian Kings and even as late as 1765 William Blackstone writing in Vol. 1, Commentaries on the Laws of England considered that the King still retained absolute personal command of the armed forces:

THE king is considered, in the next place, as the generalissimo, or the first in military command, within the kingdom. The great end of society is to protect the weakness of individuals by the united strength of the community: and the principal use of government is to direct that united strength in the best and most effectual manner, to answer the end proposed. Monarchical government is allowed to be the fittest of any for this purpose: it follows therefore, from the very end of its institution, that in a monarchy the military power must be trusted in the hands of the prince. IN this capacity therefore, of general of the kingdom, the king has the sole power of raising and regulating fleets and armies. Of the manner in which they are raised and regulated I shall speak more, when I come to consider the military state. We are now only to consider the prerogative of enlisting and of governing them: which indeed was disputed and claimed, contrary to all reason and precedent, by the long parliament of king Charles I; but, upon the restoration of his son, was solemnly declared by the statute 13 Car. II. c. 6. to be in the king alone: for that the sole supreme government and command of the militia within all his majesty's realms and dominions, and of all forces by sea and land, and of all sorts and places of strength, ever was and is the undoubted right of his majesty, and his royal predecessors, kings and queens of England; and that both or either house of parliament cannot, nor ought to, pretend to the same.

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.