Mainly because they didn't want to.

The two Arab governments in charge of the West Bank and Gaza took two different approaches to how they administered the territory, but both had grander ambitions. So keeping the West Bank and Gaza part of their own states fed into those grander ambitions. I'll split it up by country.

Jordan

Jordan was the only one who formally annexed the West Bank, a move which only the British, Iraq, and Pakistan. Side note: Iraq was ruled by the Jordanian king's second cousin, in a strange turn of coincidence thanks to the British.

When Jordan initially considered entering the 1948 war, it actually had to go back on an agreement. You see, King Abdullah had been in secret negotiations with the Jewish Agency (Jewish leadership in the Mandate) for two years, trying to convince them to accept autonomous rights within the Jordanian kingdom, which would annex the Mandate. The Jewish Agency insisted on independence, and in 1947 he gave up trying to convince them. In November 1947, he said, and I'm paraphrasing here, "At least let me have eastern Palestine". The Jewish Agency agreed (Golda Meir, later Israel's first woman prime minister, was the negotiator), and noted that they would have one less enemy in a war that seemed bout to erupt. There were no further meetings until May 1948, just 5 days before the Arab states invaded and at a time when the Jewish leadership had all but won a civil war against the Palestinian leaders. There, Abdullah once again offered autonomy within a Jordanian state, which Meir refused. He claimed that now, his hands were tied: the Arab states were in a fury, and he could not avoid going to war with them. This would be the reluctance Jordan displayed quite often, in reality, with going to war with Israel.

So why did Jordan want it all so badly, and why was it willing to accept only the eastern half (which was under the partition plan, essentially, so more than the West Bank today, but still). Part of it was his grander schemes: Abdullah wanted to create a greater Hashemite kingdom, by combining the Mandate of Palestine with Iraq and Jordan, a sort of non-contiguous federation.

Jordan treated the Palestinians to a round of almost indoctrination. Palestinians were given Jordanian citizenship and the right to vote in elections (before they had citizenship, even), and put through a Jordanian system of education. Jordan was afraid: including Palestinian refugees in the West Bank, Jordan's population became 70% Palestinian as a result of the war. This would lead to the eventual assassination of King Abdullah of Jordan in 1951, by supporters of the Palestinian Husseini family, rivals of the family Abdullah had favored in Palestinian society. Jordan attempted to teach Palestinians Jordanian history, and make them believe they were Jordanian, to extinguish Palestinian identity among them and cement their hold on the West Bank. This did not work, of course, but it was the goal of Jordan for quite some time, and something they didn't give up on for decades (even after losing the West Bank).

Egypt

The Egyptians also sought land from the Mandate, as you may have guessed. They were also, however, much less open to eventual peace or secret negotiations than Abdullah was, though some were indeed launched after the 1948 war (which went nowhere, obviously).

Egypt was a rival of Jordan, as King Farouk and King Abdullah viewed each other as competitors for Palestine. Farouk suspected that Abdullah was trying to take the West Bank without so much as lifting a finger for war, and he was unwilling to allow this to happen easily (hence the pressure Abdullah felt to go to war). The Egyptians changed their war plan specifically to try and stymie this, going from a single pronged attack to two-prongs, one of which would aim to go through the southern West Bank and towards Jerusalem, taking the land from Jordan's clutches. It was rivalry and greed, in essence, that led to the Egyptians and Jordanians both ignoring the wishes of the Palestinians.

Egypt reacted differently from Jordan, choosing to erect a mock-government in Gaza to pretend it was granting some semblance of sovereignty. Egypt pressed for the Arab League to endorse the creation of the "All-Palestine Government", led by Hajj Amin al-Husseini, who had been a prominent Palestinian leader but later fell out of favor and lost influence among the people due to his exile. The Jordanians convened the "First Palestinian Congress" in response to denounce the government, just to give you an idea of how petty the rivalry was. By October 6, after declaring the Government on September 22, Hajj Amin was bundled back to Cairo and his "ministers" soon followed. The Egyptians left the All Palestine Government on paper as an existing entity, but it had virtually no sovereignty, and Palestinians were excluded. It printed 14,000 Palestinian passports as its only achievement, none of which were recognized or useful in any way. In 1959, Nasser (well after the overthrow of King Farouk) disbanded the All Palestine Government, which had only existed in name regardless for almost a decade.

There is one more reason Egypt may have kept the Palestinians marginalized. While Jordan sought to consolidate its gains and was reluctant to go to war, Egypt was constantly aware that of Jordan's failure to convince the Palestinians that they were Jordanian. In short, it worried as well that Egypt would fall prey to Palestinian demands, so they conveniently did what the other states often did as well: blamed Zionism. They talked often and paid lip service to "Arab unity" and "removing Israel", but none felt confident they could do it. Abdullah in particular seemed willing to avoid war entirely and simply make peace, held back by other states like Egypt who were lukewarm at best to the idea of peace. But by constantly keeping the focus on Israel, the Arab states could not only distract from internal problems, but also give the Palestinians something else to focus on. Each Arab state, leading up to 1967, had different strategies for how to deal with the problem, some of which began to conflict with the rise of groups like Fatah, who carried out attacks on Israel themselves that were not necessarily wanted by the Arab states. Each Arab state tried to mollify its refugees and direct their attention at Israel.