Why did India end up tilting" to the Soviet pole during the Cold Wars, while Pakistan was pro- US pole? And to what degree was India pro Soviet?
India never joined the Soviet pole or the Soviet bloc though India and the Soviet Union (USSR) enjoyed a strong strategic, military, economic and diplomatic relationship.
After independence both India and Pakistan were backward and needed support from stronger nations. Pakistan joined the Western bloc by signing SEATO and CENTO. India developed closed ties with Soviet but remained Non-Aligned in the Cold War.
The reasons for strong Indo-Soviet ties despite NAM:
India and USSR signed The Indo-Soviet Treaty of Peace, Friendship and Co-operation on 9 August 1971.
The only 'military' related clauses in The Indo-Soviet Treaty of Peace, Friendship and Co-operation treaty are:
Both of these clauses ultimately relates to peace and non-aggression against each other.
In some of the incidents, India had also shared good relations not only with the USSR but also with United States during the Cold War.
In initial years of independence, due to poor harvest, India asked for free food from US. US made the donations and from 1947–1959 the U.S. provided $1.7 billion in gifts, including $931 million in food.
During the Chinese invasion of 1962, even in the midst of the Cuban Missile Crisis, United States was willing to help India. There were plans to send the USS Kitty Hawk aircraft carrier to the Bay of Bengal to support India.
UPDATE 17th August '13 :
Very recently declassified CIA files shows that Pundit Nehru gave permission to the US to use Indian airbase Charbatia in Orrisa state to spy on China after 1962 conflict. US was also going to use this base for spying on USSR. From today's (17th August '13)The Indian Express
So, India did not actually join the Soviet Bloc or US Bloc. As also there was no formal agreement like NATO, CENTO, SEATO or Warsaw signed by India unlike what the Pakistan did. India was just pursing its national interests, without compromising its Non-Alignment principles, with more inclinations towards Soviets than the US.
India didn't "officially" join the Soviet bloc in the sense of being a signatory to the Warsaw Pact in Eastern Europe.
India DID unofficially "align" with the Soviet Union in Asia.
This was to create a counterweight to the Chinese-Pakistani "understanding" that originated in 1962 over the Kashmir (China invaded India on Pakistan's behalf). The need for some sort of "understanding" with the Soviet Union grew further as a result of the U.S. rapprochements with Pakistan, and with China, in 1972.
India was "socialist" and therefore somewhat ideologically compatible with the Soviet Union. But mainly it was a case of "the enemy of my enemy is my friend." They actually had few interests in common to be truly "allied."