Can people living in the USA, Canada, UK, Australia, and New Zealand be considered the same/one nation?

Not now, and not for a while. But up until the ’50s, maybe: Australians certainly did define themselves as being the same nation as the UK. Australians were rather upset when the UK started using what had been the term for their common nation, to refer exclusively to the UK.

That term btw was not English. It was British subject. Which up until the ’50s means “white Anglo/Celt living in the Commonwealth”.