Have you set up the new hosting account with Hosting UK? If not then you or your web designer needs to do that.
First of all make sure you have a back up of your website from your web designer. Once the new hosting account is set up you need upload your website to the new hosting account (or get your web designer to do it) and ask Hosting UK for the nameserver and registrar tag so you can login to 123reg and initiate the domain transfer. In the 123reg control panel go to the domain name section and look for change nameserver. Change the nameservers first NOT the registrar tag. When you have changed the nameservers wait 48 hours and check the Whois to see the nameservers have indeed changed. You can check whois for:
'.com' at http://www.internic.net/whois.html
.co.uk' at http://www.nominet.org.uk/other/whois
Once you can see the nameservers are set to Hosting UK you can then go back to 123reg and alter the registrar tag. Once this is done send Hosting UK a e-mail to let them know this has all been done.
Your domain name should be sorted within 48 hours if you do this correctly. Although I would not advise you to carry out this unless you are completely confident. Ask your web designer - they really should sort this out for you. Or ask Hosting UK - they may help you if you have set up hosting with them.
If the domain is definitely in your name (check the whois as mentioned above) and you have no luck with sorting it out like that the last resort is to go to internic:
http://www.internic.net/problem_reports/p3.html
or nominet:
http://www.nominet.org.uk/disputes/