Under the Companies Act 1985 [and later additions] only limited companies need to be registered with Companies House. Sole traders and private partnerships do not need to register.
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You do obviously need to register yourself self-employed and notify the Inland Revenue and National Insurance Contributions Agency. You may apply for an exemption from additional NI contributions if you are currently full-time employed and running your self-employed staus in your own time - your earning must be less than £4095 for this exemption to apply.
VAT is a different kettle of fish. The VAT threshold is currently £56,000. You can voluntarily register even if you fall below that amount.