Why didn't Pakistan try to wrest Kashmir from India in 1962 but tried to do so in 1965?

1. USA and UK blocked that kind of effort¹. Because, they considered India as a good bulwark against the spread of communism.

Then Pakistan President Ayub Khan told Kennedy that he wanted “compensation” from India in Kashmir for Pakistan’s neutrality during the war. Kennedy made clear to Ayub that no such compensation would be tolerated, and that Pakistani intervention in the war in the Himalayas would be seen by Washington as a hostile act.

3. Pakistan and China didn't have any significant relationship. Ayub Khan signed Sino-Pak treaty after of Sino-Indian War of 1962². Before that, neither India nor Pakistan knew that China could be a friend of Pakistan against India. Wikipedia says:

It is the popular perception that the catch phrase of India's diplomacy with China in the 1950s was Hindi-Chini bhai-bhai, which means, in Hindi, "Indians and Chinese are brothers" While VK Krishna Menon was the Defence Minister in 1958, Nehru had privately told G. Parthasarathi the Indian envoy to China to send all communications directly to him bypassing Menon, due to his communist background and sympathy towards China.

Which means, during the 50's and early 60's India never considered China as a threat.

According to this link:

...... It was during the Non Alignment Movement that the friendship started converting into hostility between India and China. America started giving India more importance after the change of government in 1961. …… The defense pacts of USA with India were also perceived as a threat to its security by Pakistan. …… In 1959 the government of Pakistan offered to negotiate on the undefined boundary with China. The relations started improving when China responded after Pakistan voted for its right to be in the United Nations. The Sino-India war in 1962 also played a vital role in the improvement of relation between Pakistan and China as Pakistan’s sympathies were with China. ……

which means, before 1962's Sino-India war, Ayub Khan was either not in good terms with China, or never considered making friends with China.

4. According to Pakistani sources, Indian Army were the first to cross the border to go into Pakistan^{3, 4}.

There is no mystery here at all. Pakistan was a US ally. The US backed India in its conflict with China. Had Pakistan tried anything, the US would have severely penalised it.

What changed was Nehru's attitude to Kashmir and the reemergence of Sheikh Abdullah as a political force. It genuinely appeared to many observers that Nehru was preparing for some big concession on the Valley. His death completely changed the political picture in India and the Pakistanis believed they needed to move quickly while the Indians were in disarray. Furthermore, the infamous 'hazratbal incident' had created the impression that the Kashmiris might welcome the Pakistani Army- as a matter of fact, this did not happen.

Another political factor usually quoted is the impact of Fatimah Jinnah's campaign in which she accused Ayub of having sold out to the Hindus over the Indus waters. This wasn't true but it showed that America was being viewed by suspicion despite its genuine help to Pakistan. A complicating factor was that Ayub's attempt to modernise Islam with the help of Prof. Fazlur Rahman Malik had stirred up a hornet's nest of sectarian rivalry. Ayub thought that winning a war would make him a 'Ghazi' and give him authority over the Ulema. There were also some military miscalculations and failures of Intelligence. Still, had there been a popular uprising in the Kashmir Valley and also if certain key operations had not been mismanaged, Ayub could have claimed it as a victory.

Turning to the Indian side, three factors which no one talks about should be borne in mind

1) appearing to lose calamitously to China was very good for India. It split the indigenous Communists and, unexpectedly, killed off Dravidian separatism. It turned out that Indian Nationalism was a stronger force than had been previously imagined. However, losing to Pakistan would have been an intolerable narcissistic injury. For a start it would have led to ethnic cleaning on a massive scale within India. Secondly, India would have focused on an infantry led assault across the Punjab and Sindh border rendering Lahore uninhabitable. Thus Pakistan would lose more by winning. As it was, the Indians got worried about the 'Siliguri gap' which some loudmouth Pak generals were talking about- thus they welcomed the opportunity to dismember the East Wing- i.e. Bangladesh, though this was a blessing in disguise for Pakistan.

2) The death of Nehru meant that America saw India as potentially coming into the Western Camp on a wholesale basis. Since the regime change in Baghdad and the rise of Nasserism, India had more weight in the Middle East than some General in Pakistan. However, the unexpected death of Lal Bahadur Shastri and divisions withing the Congress 'Syndicate' meant that India could not become a US ally. Post '65, Pakistani diplomats did some pretty nifty footwork- while the Indian diplomats and politicians fell flat on their faces- so that, by the time Nixon was in the White House, America had given up on India and vice versa. Still, America could not commit to Pakistan against India because India simply mattered so much more. Anyway, Indira Gandhi signed a Defence Pact with the Soviets, so there was a limit to what America could do.

3) India, like Ceylon, had intially been distrustful of the Army. However, unlike Bureaucrats, Soldiers have useful skills. Post '62, the new consensus was that the Army should be looked after and insulated from the politicians. This was a sound policy. India developed a professional Army and was less adventurist than its neighbour as a result.

'65 was by no means a disaster for Pakistan or, indeed, for Ayub personally. The people of Punjab saw they were vulnerable and supported the men in khaki. Esprit de corps and fighting spirit (josh) tended to increase not fall. Ultimately Ayub came to be seen as standing in the way of reform within the Army. However, it was mass disaffection- in part due to economic progress made under Ayub- which brought him down. However, his successor was worse so it was only after the Bangladesh war that the Pakistani Army started to assume its modern form.