What percentage of the world is asleep at any one time, and does this figure greatly change throughout the day?

It will change greatly throughout a 24 hour period.
As China and India have around a third of the world population between them, when it is night there there will be more people asleep than any other time.
This site gives population density, it will clearly show as night moves round the world form Indo-China fewer people will be asleep.

https://en.wikipedia.org/wiki/Population_density

What that doesn't take into account is the normal working hours of the people. As India provides most of the UK companies call centres these days they must sleep at odd hours compared to a normal working pattern

For every person in an Indian call center working on a British or American time zone, there are hundreds (if not thousands) who are living and working in Indian daylight. There are people working in every country 24/7, but most of the population of any country sleeps during the night.

Also, the Pacific takes up a good number of time zones, and maybe a few percentage points of the world's population lives there.