Before Nominet a very few select groups got one:
Parliament, MOD, British Library, National Library of Scotland and the worlds most powerful nuclear fusion reactor based in Oxford.
I only know of one private company that got one: A web designer in Newcastle who had the cheek to call himself The W3 Organisation and registered org . UK
Today it is a one page site his wife has just sold.
while
ac.uk Higher and further education and research institutions
gov.uk National, regional, and local government bodies and agencies
mod.uk Military and related purposes
nhs.uk National Health Service
police.uk Police forces
.nic.uk Network use only
.sch.uk Schools
do exist, but arn't looked after by Nominet, plus oddities like parliament.uk