What are some of the British Empire's holdings in Asia, South Pacific, or Western Hemisphere?

Mostly islands:
Isle of man, Orkney islands, South Orkney
Tristian de cuna, st helena, Falklands, Sandwich, Georgia, and that big US bomber base in the Indian Ocean-Diego Garcia (The US leases it from the Brit's-thank you!). I think it part of a territory called "Indian Ocean Territory".
Oh yeah-duh- British Virgin Islands....

Not the USofA since 1776-eh?

Not bad for US (Missouri) public school level geography?

Ummmmm ... even in Missouri, they teach that the Isle of Man and the Orkneys are not in Asia, the South Pacific, or the Western Hemisphere.

The Falklands and St. Helena are in the Atlantic ocean, as are South Sandwich Island, South Georgia Island, and Tristan da Cunha.

In the areas you specified, the British only retain the Pitcairn Islands in the Pacific [only Pitcairn is inhabited - the other three are uninhabited atolls] and the British Indian Ocean Territory, which includes Diego Garcia.