Your domain name and your website hosting are related but separate. If a new provider is going to manage the domain, the registration needs to be transferred or moved to that provider. The exact process depends on the domain extension. UK domains and generic domains can use different transfer mechanisms. If you already own a domain, make sure you can access the account where it is currently managed before starting a move. If you have lost the login, you generally need to recover access with the current provider first. Moving the registration does not automatically mean that email or DNS settings should be changed immediately, so a careful provider will preserve existing services during the transfer.
What to look for
Choose a website provider based on the complete service, how clearly the responsibilities are explained and whether the package fits the way your business actually works. A good website should be easy for customers to understand and easy for the business owner to keep current.
Keep the commercial details clear
Before signing up, check the price, minimum term, page/email limits, domain arrangements, support and cancellation position. If SEO is included, understand that meaningful SEO is ongoing work rather than a guaranteed position in Google.