First, I am not a business lawyer, or tax attorney. The following are my understandings from being in business, do not rely on this as legal advice, and you should consult with a business lawyer or tax attorney in your jurisdiction.
As a general rule, No, my understanding is that writers do not need a business account, or ANY account, for that matter. I will discuss exceptions below.
For the purpose of taxation (at least in the USA, and perhaps UK), what writers need is receipts, invoices, and proof they paid for stuff. For expenses where you don't get a receipt, the proof can be a line item on a personal credit card, or a cleared check on personal checking account statement. What the tax people are looking for is proof of what you claim you spent in the pursuit of profit. Whether the expenses are enough to bother itemizing and deducting from income is a matter for a tax professional.
When you need a Business Account
It is generally a good idea to form an LLC (Limited Liability Company) or some kind of corporation to shield your personal finances from business lawsuits. When that is done, any lawsuits (for libel, copyright infringement, plagiary, disputes over contracts) should go against the business entity that owns the property (books, art, intellectual property, copyrights, trademarks, and liquid assets like money). That is why they are called "Limited Liability", your personal assets and the business assets are distinct.
However, in the USA at least, if you have a history of mixing your personal finances with the LLC, somebody suing you can claim they are one and the same. That is called "piercing the veil", and liability is no longer limited AT ALL.
So if you have an LLC, you must be meticulous about not paying its bills from your personal credit cards or checks, or in cash out of your pocket, and vice versa: You can't write a company check to make your car payment. Likewise you can't buy personal things, claim they were for the company, and use them at home. The company can buy itself a copier, but it can't buy you groceries for your home. (There are legitimate reasons for an LLC to buy groceries, many restaurants and food trucks are LLCs, but tax men know the difference between that and buying home groceries, or paying your apartment rent.)
This is true even if you own 100% of the LLC, if you want money out of it, then it pays you, abides by any withholding and reporting tax laws for that, then it becomes your personal income with which you can buy groceries.
The same holds in the opposite direction, like if the company needs to pay for cover art or something and doesn't have enough money. You don't just write a personal check; there is a process by which you give money (or property) to the company, and you have to follow it.
JUST having a "business account" or "business only credit card" does not protect you, you must actually form the LLC or corporation. Without that, you are just engaging in a convenience to organize your finances, not protecting yourself from liability. There are on-line providers that ARE lawyers and help people set up LLCs for a few hundred dollars. Google "Form an LLC" for recommendations. I believe the equivalent idea in the UK is "Ltd", a private limited company.
Prices vary, I have set up multiple LLCs each for around $300 US, but some of these providers pile on extras you don't need, I have heard of people spending $1500 on one of these providers to get one LLC set up. Buyer Beware. And if you do it, don't mix your personal and business finances except by the approved routes.