Question : What are the requirements to register my small business as a foreigner in London?

If by small business, you mean a company, then all that is required is to register the company at Companies House, to include a legal address known as the registered office, then to register for corporation tax. VAT registration is not essential unless the annual turnover will be £81,000 or more. Payroll registration is only necessary if employees are to be employed. A bank account will be required, and this will involve satisfying the Bank’s due diligence inquiries about the directors and shareholders.