In the Great Barrier Reef, 100 miles east of Queensland, Australia.
Dolphin Island isn't so oceanic as I'd thought - it's pretty close (at least by Pacific Ocean standards of closeness) to the Australian continent.
From Chapter 5:
"Just a minute," pleaded Johnny. "Please tell me where I am. Is this Australia?"
The nurse took her time in ...
Early 21st century, probably the first half of the 2010s.
There's a few pieces of information scattered through the novel which allow us to pin down the approximate time period in which it must be set.
In Chapter 1, the opening paragraph of the story sets the scene in the then-distant 21st century:
Johnny Clinton was sleeping when the hovership raced down ...