While more can always be said, you may find this previous answer by u/drylaw helpful regarding Spanish America. In short, Spanish colonies tended to be in more densely populated regions and relied more on exploiting indigenous labor, while British colonies were often located in less densely populated regions and relied more on taking indigenous land for their own use.

I’ll try to add some more on the British colonies and the specific factors that led up to removal, at least in the United States. (I will not focus on Australia, and I will not focus as much on Canada - I am not knowledgeable enough to comment on the specifics of each of those). One important factor is immigration - British settler colonies were magnets for immigration in the 19th century, leading to a massive influx of Europeans and making the population “whiter”. By contrast, most Spanish colonies saw substantially less immigration - fewer Spaniards came over in the first place, and after independence Latin America was not as attractive to immigrants as Northern America (with exceptions - Argentina or Uruguay, for instance, attracted large numbers of immigrants and also possess a smaller indigenous or mestizo population than countries like Mexico or Peru.)

All this said, British settlers weren’t chasing indigenous peoples across the Appalachians as soon as they got across the Atlantic. Removal, as a process, took a long time to develop into the full brutal form it would take in the 19th century. First it should be noted that there was a noteworthy temporal gap between the first Spanish colonies and the first British ones. In fact, large portions of what is today the United States had been explored and charted by the expeditions of Hernando de Soto, Francisco Vasquez de Coronado, and Juan Cabrillo, among others, long before the British attempted to land at Jamestown in 1607. By that point, the Spanish Empire covered vast swaths of the Americas, and the racial and political dynamics that would come to define Spanish America had already started to take shape. The British were not ignorant of this - Spain’s actions in the Americas were well-known, and in fact Spain had acquired a rather unsavory reputation in much of Europe for its atrocities towards Natives. This would take shape as the Black Legend - the assertion that Spain, as an intrinsic result of Catholicism or “Spanishness”, had been uniquely and horrifically brutal towards indigenous peoples. It wasn’t necessarily wrong about the atrocities - Spain performed many brutal, even genocidal actions in its rule over the Americas. But this was not something uniquely Spanish or Catholic, and it would be perversely twisted by the British to justify atrocities and genocides of an equal or greater magnitude than the Spanish.

The Black Legend informed much of early British colonialism. The British - viewing themselves as more enlightened than the Catholic Spaniards - had to be different, and their colonialism therefore had to be better. The other crucial, and inarguably more important, prejudice at play was the British view of indigenous peoples. It was clear to the 17th-century British, and to 17th-century Europeans more broadly, that their cultures were superior to those of Native Americans. The view of a continent of ungoverned savages was widespread; Thomas Hobbes refers to North America as exemplifying the violent “state of nature” he decries in The Leviathan. Indigenous agriculture was derided as not living up to European standards, and this then taken to mean that Native Americans did not work the land they lived on. Following John Locke and other philosophers of the time, this meant that indigenous peoples did not own the land they lived on. It was, in theory, free for the taking. In practice, this would be much more complicated, not least because Native Americans had no desire to surrender their land.

It was in this context that the first British colonies in what is now the US were established at Jamestown in 1607 and Plymouth in 1620. While there were great differences between the settlers of each, both shared an ideology that held English culture to be superior over both Native and Spanish culture alike. Especially among the Puritans at Plymouth, the fear that they would “decline” and become like their indigenous neighbors was a major fear of the early settlers. It was a not-uncommon belief in Europe at the time that living in the Americas could transform even a civilized European into a “savage”, just by virtue of how the continent was. British settlers fought vigorously against this feared backsliding. The earliest settlers in the northeast built wigwams not unlike their indigenous neighbors, a practice that was quickly discouraged and fell out of the way. It is not surprising that in this climate of prejudice and fear, British settlers were not keen on mixing with indigenous peoples. And unlike the Spanish, many of the British settlers were married families, so there was not the dearth of women common among the Spanish colonists.

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