Great question!

What's important to understand is that in 1956, the British had an interesting policy in the Middle East. They were largely being displaced as the primary power in the area by the United States, and whenever the United States was not replacing them, the Soviets were. This was especially clear in Egypt, which was seen as pro-Soviet by 1956, following the Czechoslovakian arms deal of 1955. While the West had tried to keep Egypt in its orbit through the early 1950s, and the US had even pressured the British to acquiesce to Egyptian demands regarding Suez bases the British held, the decision to accept Soviet arms in 1955 instead of US arms was seen as the clear indication of Egyptian intent. This was doubled up by the decision to nationalize the Suez in July 1956.

This relationship of contention with the West, and of fading British influence, can be contrasted and compared with the British-Jordanian and British-Iraqi relationship. In the 1950s, the British were still heavily involved in training and funding the Iraqi and Jordanian militaries. Sir John Bagot Glubb, known as Glubb Pasha, was a British soldier and commander who was still in command of the Arab Legion (Jordanian Army) as late as March 1956, and he remained close with the king of Jordan even following his dismissal.

This complicated relations with Israel, because Iraq and Jordan were both still technically at war with Israel (Jordan did not formally end its war with Israel until 1994's peace treaty). The problem was, Nasser was beginning to extend his pan-Arabist propaganda, and gain influence in Jordan and Iraq. The British were still funding the Jordanian army, but this would end in 1957, another mark of slowing and fading British influence.

In the face of rising Nasserite influence on the ground, the UK had already considered sending Iraqi troops to Jordan to ensure no unrest, which Israel opposed forcefully. But while this was being contemplated, there were a variety of back-and-forth reprisal attacks between Israel and Jordan that raised tensions. On October 9, there was a raid by Jordanian soldiers that killed Israeli workers in the Sharon Plain, so Israel organized a counter-raid on October 10 that went awry. Israeli troops, having destroyed a police station near Qalqilya in the West Bank, were also ordered not to go into the town (to avoid civilians) and were ordered not to capture a Jordanian military post down the road that Ariel Sharon (the troop commander) wanted to take. Yet after planning an ambush of Jordanian soldiers as part of their mission, Sharon took heavier than expected casualties, and had to break the military post to escape, contrary to Dayan's orders. The raid cost 18 IDF paratroopers, but Sharon managed to get his troops back to Israel, which was crucial; neither side had to deploy their air force to attack (or defend against attack) the paratroopers still in the West Bank, and could avoid further escalation.

At the same time, Israel was propositioned by the French, in what would become the Suez attack that turned into the Suez Crisis. The British were in favor of the proposal generally, as long as the Israeli action did not appear prearranged. In fact, it seemed almost like the British wanted to avoid being held to any arrangement at all, and the British leadership was furious when their representative at these meetings signed quite detailed minutes on the meeting, meant to record the agreement. Israel, on the other hand, was jubilant that hard proof existed of the Suez planning; Israel had feared that the British wanted Israel to invade the Sinai so that it could not oppose British plans to deploy the Iraqi army in Jordan.

By now you should have a decent understanding of what the Israeli-British mindset was. The Israelis and British did not have a deep relationship or trust in one another, less than 10 years since the end of the British occupation of what became Israel.

The Suez Crisis and prioritization of it kept the British and Israelis from any hardening of conflict. But in theory, they were close to blows at one point, during Sharon's raid. You see, when the Jordanians realized the Israelis were conducting the raid, they actually reached out to the British for aid, asking them to "extend help" from their airbase in Cyprus. Prime Minister Anthony Eden actually said that his planes "were about to take off" in response to the request, which could have led to a diplomatic standoff and potentially a war.

This episode is hardly reported on, but Dayan fretted over the possibility. Dayan actually requested that the air force ready itself, in case the paratroopers were not rescued by dawn from the West Bank.

So, why would the British have gotten involved at the Jordanian request?

For one, they still viewed themselves as a great power, and wanted to preserve influence in Jordan. They had no special friendship with Israel, and were still bitter about the attacks on British troops and personnel from the Mandate era. Given that they were still funding the army that the Israelis had just raided and attacked, their involvement (or at least a show of force) was certainly possible.

The other crucial factor here is that the British had a defense treaty with Jordan. They could avoid war for plausibly deniable actions, but not something as public as sixty surrounded Israeli troops, if they were not extricated immediately. In fact, near dawn, the British charge d'affaires knocked on Ben-Gurion's door on October 10 and told him that he wanted "to remind [Ben-Gurion] of [the British] defense treaty with Jordan. If [Israel] attack[s] Jordan, [the British] will intervene". This was before the attack, by the way, because of rumors of a planned Israeli response to the Jordanian raid the day before. The British-Jordanian defense treaty, which codified the British funding of the Jordanian army as well, would be replaced not a year later, but at the time was still considered to be a binding obligation. It was, in fact, a treaty that had been in effect since the creation of Transjordan through it, in 1946. Article 5 of the treaty established that if either side became embroiled in hostilities, the other would come to their aid in collective defense, and even said they should consult in the case of imminent hostilities. Well, to be clear, those were the provisions in the Treaty of London concluded in 1946. Technically, this had been replaced by the Anglo-Jordanian Treaty of Alliance of 1948. I tell ya, this one was not easy to find online, but it can be found if you look hard enough.

The 1948 treaty switched around the collective defense articles, but Article 3 effectively said the same thing: either side must come to the aid of the other in the case of hostilities, and must consult even if hostilities just appear imminent. So the British were, indeed, obligated to come to the Jordanian defense, and a too-public Israeli raid could cause Jordan to request British aid as it did, and the British would be treaty-bound to respond with assistance. And as I noted, this was something the British themselves warned the Israelis of.

How close were they, truly, to war? While the British were readying planes, it seems unlikely to me that the British would have been necessary to attack those commandos. Instead, it seems more likely that either the Jordanians would have captured/killed them, or the Israelis extricated them, before the British planes arrived on the scene. Furthermore, in the event of a war with Jordan, so long as Israel did not formally invade Jordan, it would not necessarily have triggered British involvement. And Israel did not truly seek a war with Jordan; part of its counter-raids into Jordan were to divert attention from the true Israeli goal, which was to consider the toppling of Nasser, a goal Israel had sought even before the British approached Israel with the Suez plan. And since the French and Israelis had begun discussing the Suez plan in June or July 1956, and the British had begun planning an invasion of Egypt in July 1956, it seems unlikely that the sides would have wagered their plans for Egypt on a war over Jordan. Still, it was a real quandary because had the British been sucked into taking sides on the Israeli raid on October 10, it would have cost them significant face or forced them to attack Israeli forces, which could have thrown the Suez planning into disarray, given the deal had not yet been sealed. Dayan's worry, therefore, was somewhat legitimate, particularly when he was still waiting for Sharon's return, and events began to unfold in earnest. So while unlikely, the possibility of British intervention was certainly one that the Israelis had learned to watch for and tread carefully around.