There's no requirement to register a business name if you're a sole trader except that
(a) HMRC will require you to state the name that your business trades under when you register for self-assessment ; and
(b) you can (optionally) register your business name as a trade mark if you want to prevent others from using it.
Remember that any 'official' paperwork you provide to others (such as invoices for your products or services) must show your
own name as well as that of your business. (i.e. you can't just put 'Demeter Gardening Services' on your paperwork, with no mention of your own name. You need to state something along the lines of 'Fred Bloggs, t/a Demeter Gardening Services).
https://www.gov.uk/set-up-sole-trader