Did Nepal was part of India?

Based on the wording of question itself, you have to know the meaning of ‘India’ firstly. The question misleads as there was India in the history as is now. Meaning of India before 1857, 1857-1935, 1935–1947, 1947–1971 and 1971 till now is different. You can combine the concept after 1947 till date too. To shorten the answer, we may take the concept of India at 1935. Taking modern geographic names, in 1935, modern Pakistan, some major part of modern India, modern Bangladesh and modern Myanmar was British India. At that time there were around 600 other sovereign states or British dominated independent States. For the whole reason including of above all, plus other independent or separately dominated states like Cylon was collectively named as India.

     Read These For Historical Periods Of Nepal

    Nepal was actually never part of India for a long period or continuous period and if it was, not all of Nepal was part of India. As for historical periods before Mughal came to India, only four Indian empires had some kind of infulence/relationship with Nepal. The first one was the Shakya Janapada. Buddha was a Shakya prince. It ruled the region today known as Terai in Nepal. It 's territories most probably spanned the northern reaches of Gangetic plains in all probability. The second one was Maurya empire. Under Maurya emperors, southern Nepal was under Magadha's rule. As a symbol of that bygone era, one Ashokan Pillar is still standing in Lumbini, birthplace of Buddha.
The third one was Guptas. Guptas were either overlord of Nepali kings or in matrimonial alliance with them. Samudragupta proudly proclaimed himself as Licchavidauhitra i.e. son of the Licchavi's daughter. So most probably the matrimonal alliance theory is more correct.
The fourth and last one was Chalukyas. They during conquest of Northern Plains had invaded South Nepal. But their occupation was brief and soon Chalukyan power waned.
These were the instances when Nepal was someway under political control of Indian kings or emperors. It os evident from the timeline that the political control was not for long or continuous.

Jang Bahadur visited Great Britain during the time of Queen Victoria. British newspapers respectfully spelt his name as Indian Princes, who were first highest political leader from Indian geography visited them. This means around 1863, meaning of India was territory of South Asia not any state.

If we read Kautilya’s, there were many (exact number unknown) Indian countries named in his works. Without hesitation, he wrote them as Indian collectively. This means at that time 200+ BC, meaning of India was collective area not a name of state as now. In 1947 British left their political power saying their control divided into two dominions. During independence from them, India and Pakistan formed and both included other territories around 675(?). Political name of two Nations became India and Pakistan. Thenafter meaning of India limited into name of a nation rather than geographical region.

Before 1947 (mainly before 1935), there was no concern of India (Bharat), they arbitrarily used either Bharat or Bharatvarsa. Then after, neighbours recognised India (Bharat) as a nation and their territory as Bharatvarsa. In 1985, SAARC has established and historical name of Bharatvarsa also going to be disappeared.

In light of above, Nepal (it is also combination of many native states) never controlled by either historical Indian States or its current territory. But one thing we should not forget is that, the territory controlled by Nepali States has narrowed many times.