It looks like you own both domains (they’re both registered with 123.reg), so changing to it will really just require you to direct your incoming traffic to the server. Right now, the two different addresses point to two different Web servers:
marvin:~ waltd$ host markheeler-kitchens.co.uk
markheeler-kitchens.co.uk has address 188.8.131.52
marvin:~ waltd$ host markheeler.com
markheeler.com has address 184.108.40.206
Those nn.nn.nn.nn numbers are the true names of Internet-connected servers. DNS exists as a kind of phone book, where you can put in a name and get a number (which is really the only way to connect to another machine).
It is common, but not required, to have your DNS handled by the same company that you pay to register your domain name. So 123.reg would probably have the controls to let you point markheeler.com and www.markheeler.com to 220.127.116.11.
You will want an A record pointed to www, and a CNAME record pointing from the bare domain to the www. (Counter to what you’d expect from their names, CNAMEs, or Canonical Names, are like aliases to a real machine name, and A’s are true addresses.) Depending on the control panel exposed to you at 123.reg, you may see a setting to point *.markheeler.com or @.markheeler.com to an address. This is a shortcut for the above configuration.
It is best if there is only one A record pointed at one IP address, so if you want to continue getting traffic from www.markheeler-kitchens.co.uk to 18.104.22.168 (the current setup), you should also change the existing A record that binds those two to a CNAME pointed at www.markheeler.com. CNAME aliases can jump the fence like this, pointing from a name to a name, unlike A records, which always point from a name to a (numerical) address.
One last caveat, related to search engine optimization. Google (and others who follow them or re-use their findings) tend to mark down sites that have the exact same content addressable by two different URLs. You may want to replace your current content at www.markheeler-kitchens.co.uk with a bulk redirect; using either .htaccess or Apache configuration to set up a 301 redirect. When a visitor comes to www.makheeler-kitchens.co.uk/page-name.html, they will be redirected to Bespoke Kitchens in West Sussex | Luxury Handmade Furniture in London, and their browser will also receive a 301 header indicating that this move is permanent. That header is what Google will use to update their index, and eventually, all incoming search engine traffic will go to the new server, not the old one. There are a lot of threads about configuring that detail here, just search for ‘permanent redirect’ in the Web view of this list.
You’ll need to update your FTP settings to match the new hostname, if you get rid of the old one. You’ll also need to adjust the Apache (Web server) settings at your hosting provider so that the Web server knows to respond to requests for the new name. I forgot to add that part in my other reply.