As I understand it, in the UK it is entirely possible to trade as a business without a VAT number and without Companies House allocating you a number.
I have my own limited company that is VAT registered, and in the past I've also run a business as an unregistered sole trader so I know the ropes.
In the UK, you
have to register for VAT if your turnover is, or is reasonably likely to be
more than £85,000 in any year (full set of thresholds here:
https://www.gov.uk/vat-registration-thresholds). Below that threshold you
do not have to register for VAT, but you
may register if you so choose. Once you have registered you are
required to charge VAT, supply quarterly VAT returns, keep suitable formal records and obviously pay over VAT to the government that you have collected over and above the VAT that you have paid out to suppliers, so the advantage of registering and claiming input tax is offset by the records keeping and reporting requirements. Thus many people who
may register but are not
required to do so choose not to register. Most people will acquire the addition expense of paying a professional accountant if they opt to or are required to register for VAT.
Many, many, sole traders and partnerships run a business in the UK without forming a corporation, without registering for VAT and without being required to hold a general 'business license' as is common in the more bureaucratic parts of the world. It's very common for tradesmen like electricians and plumbers to operate a business on this basis.
There used to be an intermediate ground between being a sole trader/partnership and running a registered company of having a
Registered Business Name, but that was abolished some years ago in 1982. You only
had to have a registered name if you wanted to use a trading name other than your own, other than that it was entirely optional. I used to have a Registered Business Name back in the 70s because it allowed you to open a bank account in that name and avoid having to refer to yourself as "Jerry Mouse trading as Mousetrap Consultants" on your letterheads, invoices etc.
One requirement that
is placed on electricians working as sole traders by the IET wiring regulations/BS 7671 - having test equipment that is traceably calibrated. Bit of a gotcha for them if they opt to buy Keysight test gear but Keysight will subsequently refuse to recognise them as legitimate businesses because they aren't VAT registered or haven't formed a limited liability company.