Firstly, I have no legal experience. As far as I believe anyone can set up a domain name if it is available, there's no stopping someone just because you have set up a company with the same name. Usually I would say the first step when planning a business is to make sure the name you want is available as a domain name.
Regarding companies house, sole traders and partnerships don't need to be registered with companies house. But that is a red herring, because as I said anyone can buy a domain name, they could buy it as an investment in the hope someone will pay big money to buy it off them.
Your only options as far as I can see is either request to buy it from the owner or change your company name. It is best to buy at least .com and .co.uk versions of your business name to stop someone else getting it.
Usually if you search for a domain name and it's taken you can see who owns it, along with contact details, so you may be able to offer to buy that way. However a person can opt out of having their owner details listed so that may not work.
Can you get a .net version or.eu? My company domain was taken but with a small tweak I managed to get something very similar. Anyone can buy whatever Domain they want, it's not linked to Companies House or anything like that