This is like asking if Julius Caesar was considered Italian, if Jesus was considered Israeli, or if Charlemagne was considered French or German. With all three examples, the modern notion of the nationstate, citizenship (jus soli, espectially), and the countries in question were so far from existing in any form we would recognize that the question itself becomes ludicrous.

Julius Caesar was Italian in the sense that he was born on the landmass of "Italia" but he didn't have a passport that modern nationstates have. "Israel" was the designation of a tribal order, not a country and certainly not a nationstate, so if Jesus was ever referred to as an "Israeli" (and I don't know if that word was ever used in his direction) it was in the context of a tribal designation, not a citizenship. The same can be said for Charlemagne, where the lines blur even more considerably. Charlemagne (as his name implies) is a French national hero. He was one of the founders of the Empire of Frankia. Frankia = France. Easy, right? Well, his capital was in Aachen (modern Germany), and the Franks of the 9th Century were also a tribal designation, not a national one, and spoke a language that was closer to Old German not Late Latin. (Ask Germans who "Charlemagne" is and you get confused looks, finally registering that you mean "Karl der Große" pointing to him being a national hero of Germany as well as France due to the Treaty of Verdun that his grandchildren agreed upon).

To take a very simple example: was Pocahontas considered American because she was born in what is now considered Virginia? No, of course not. America didn't exist when she was born. The physical landmass had only recently been referred to as "America" in the first place, and the country we know today wasn't anywhere close to existing when she died.

Whether Siddhartha Gautama was considered Nepalese or Indian faces all of the same problems.

The claim that the Buddha was Nepalese stems from the fact that he was born in Lumbini grove, which is essentially on the border between modern Nepal and India. The border itself was determined in 1816 when the British signed the Sugauly Treaty with the Gorkhas in the Anglo-Gorkha War. 1816 was almost 2400 years since the Buddha was born, so the idea that we're determining the Buddha's citizenship based on a border that didn't exist is silly.

It gets even more ridiculous when we look even closer at the history. The word "Nepal," and its derivatives "Nepali" and "Nepalese," has an unclear origin. But the modern nation state derives its origins from the migration of Gorkhas (a tribe originating from modern Rajahstan) occurred in the medieval era, with the Shah Dynasty taking power in 1559 and ascending to an absolute monarchy in 1746. Over the years increasingly placing other Gorkhas and Nepali speakers in positions of power ahead of others in a country of 123 languages and as many ethnic and religious groups. While it seems that the terms "Nepali" and "Nepalese" are traditionally interchangeable, it seems to me (and my experience with the terms) that there is some basic differentiation: essentially that "Nepali" refers to speakers of the Nepali language (who happen to also be Gorkhas, formerly an ethnic distinction, now overtaken by the title "Nepali" as an ethnicity), while "Nepalese" is more referent of a citizenship/national designation. Tenzing Norgay, for example, the Sherpa who climbed Chomolungma (Everest) with Sir Edmund Hilary, was Nepalese, but not Nepali. (Don't get in a twist though, because if this distinction exists at all, it's still not a common distinction to make.)

So was the Buddha a Nepali speaker? No absolutely not. Not any more than Julius Caesar was an Italian speaker. Was he Nepalese? Well, no, not any more than Pocahontas was American because she happened to be born on a piece of land that centuries later would come to refer to a nationstate known as "Nepal."

Ok, so was he Indian?

We face most of the same problems here. "India" refers to the Indus River Valley. A name that was partially designated by Greek geographers referring to the lands off in that direction. It's doubtful that Ancient Indians in the Buddha's locality would have referred to themselves as "Indians" in any sense of the term because they didn't live anywhere near the Indus, they lived in the Ganges River Valley. And the Kingdoms of Magadha, Kosala, and Kapilavastu (and others) of course didn't count or treat their citizens in the same way that modern nationstates treat theirs.

The word that Indians of the Buddha's day probably referred to their "country" was as Bharat. Which is still a word that Indians in their various languages apply to their country and themselves. The Indian constitution (c. 1949) refers to the country's name as both "India" and "Bharat." Of course, in English, we have traditionally used the name India (Chaucer himself refers to the nation as "Ynd" and "Ynde" c. 1399) deriving from the Greek "Sindhu." And since the Subcontinent shares a common linguistic, philosophic, and historic legacy, it's really more of a civilizational marker when we refer to the Buddha as "an Indian sage." Not a national one.

So was the Buddha Indian or Nepalese? Well was Charlemagne French or German? The answer is both yes and no.

tl;dr Siddhartha Gautama is an important and critical figure to the national epic of both Nepal and India. He was born within the current borders of Nepal and spent most of his life in the border of modern India. Borders that weren't established until thousands of years after his death, and nationstates that wouldn't exist until over a century after those borders were drawn. In narrative historical terms, both countries have a claim to him. But in strict historiological terms, he was neither. If he was one, he is both. If he is not one, he is neither.