ASAP is used quite frequently in Great Britain, although the older generation are less likely to use this abbreviation. In everyday usage over here you probably hear both 'ASAP' as many times as you would hear 'as soon as possible' Like Stencel said it is usually said A-S-A-P rather than a-sap. It is most commonly used in commands or instructions such as:
Give me a ring ASAP!
Get back to me ASAP!
I need your assignment ASAP!
(That last one I hear quite often from my uni tutor
In my experience it is also used a lot in military commands when speed is required on a task. It will most likely remain in the dictionary as an abbreviation until the drawn out version 'as soon as possible' is extinct.