The .uk TLD predates the modern DNS by a couple of years IIRC. The concept of domain names existed before the idea of distributing the namespace (prior to DNS every machine on the Internet had to have an /etc/hosts file, which contained IP address to name lookups for every other machine on the Internet, a mechanism which became impossible to maintain, hence DNS). Nominet is in charge of .uk, but doesn't run government and academic SLDs (.ac.uk, .gov.uk, etc), it does run all the others however. Nominet used to be very keen to ensure that domain name registrations were made in the "proper" second level domain, or SLD (only registered non-profit making organisations would be allowed .org.uk, for example), but largely gave up on that policy a few years ago, nowadays it's mostly a free for all (with some exceptions, such as .ltd.uk, which is restricted to limited companies, and .net.uk, which is limited to ISPs and bandwidth suppliers).
There's no technical reason for using the .co.uk SLD rather than the .uk TLD, it's purely an accident of history.
Nominet is an organisation that is in theory run by its members (I'm one of them, btw), but is in practice run by Dr Willie Black and a few of his cohorts, who make all the decisions about how to run the namespace.
-- Ian