At the end of the day, relationship between states are based on self-interests, and those states who don't follow that, are bound to suffer the consequences (of not looking after their self-interests). This does not mean other states should be harmed for promoting self-interest ofcourse.
Also by Khilafat movement here, I am referring specifically to the Khilafat/Non-cooperation movement launched by INC and other leaders at that time.
It was this mass non-coooperation movement or satygrah that was opposed by Jinnah and Muslim League leadership. Jinnah also wanted a proper settlement to the Khilafat question and he along with ML elites wanted to sent deputation to the Crown and deal with it constitutionally. They did not want any agitations or protest or any non-cooperation against the British rule.
On the other hand, the Jallianwala Bagh massacre in Punjab passing ofthe Rowlatt act to curb protests and on top of that the way despite Indian support the British has in a way reneged on the understanding that India would recieve self-rule had fuled hardliners and Khilafat question just added more fuel to the already explosive situation.
It was in this context that the Khilafat/Non-cooperation movement was launched, and is remembered as one of the high points of Hindu-Muslim cooperation against British rule in India. To try to understand the Khilafat movement in isolation with the Non-cooperation/Satyagrah movement will not allow you to see the whole picture.
By nationalist, I mean those who did not support sepratism and were against the British rule in India. Hence nationalist Indians. Keep in mind that a lot of non-muslim Indians also supported the Khilafat aims including Rajagopalachari, Lala Lajpat Rai and ofcourse Gandhi.
Yes ofcourse, I am not saying that Turkey and Pakistan have cut off relations, but the unquestioned support in Cold war and hostility to India being percieved to be in the Soviet camp is no longer there.
AKP, I would believe, will build relationships with both Pakistan and India but would avoid taking a hard stand on issues like Kashmir.
At the moment, Pakistan's biggest problem is to bring a sense of stability back in the country. Without that no country would be willing to step out on a limb except if they had a strategic interest there. Hence you see the biggest aid/donors/supporters of Pakistan being US and China both for their own strategic reasons.
The same reasons apply for Turkey trying to improve relations with China and India, two countries with which it had hardly any relations just 20 years back. Comparing the diplomatic relations of India and Turkey to about 20 years back, there has been a huge improvement and its about taking advantage of the economy and infuence of India in international foras.