ok well YOU need someone to register the domain through, set up the domain (and tell them it is already registered and you own it) - then it depends on what the domain name is for what happens next.

For .co.uk you need to get the original registrar to change the IPS tag on the domain to the new registrars value. For .com I think it has to send an email to the admin email address on the domains DNS settings.

Just register the domain with your prefferred company and follow their advice / help.

Would that take care of contact details and lawful ownership of the domain

yes - once you have secured the registration via that method you will have ownership of the domain and will be able to update the DNS information wit hthe relevant info.

Again the exact rules are TLD dependant :)