1. The European invasion of North America [but let's not get bogged down in the French and Spanish influences still apparent there] was a true invasion and, compared with pre-colonial times, there are basically no Native Americans left.
Those few who survive either manage against extraordinary pressures to preserve their own languages; or they succumbed to those pressures and now speak English because they essentially live in someone else's country.
It's not a question of whether the natives changed their language: the place is full of invaders (who still speak the languages they brought with them).
Did the Anglo-Saxons displace the natives in Britain?
2. In India, the populace was subjugated by the invading ruling class, who chose, on the whole, not to sully themselves by too close an interaction.
Did the Anglo-Saxons rule Britain?