A UK company who wishes to set up a US branch/subsidiary will want to create a business entity in the United States generally in the form of a corporation (given its ability to offer insulation from personal liability). It is important to understand that there is no such entity as a “US” corporation rather a business entity is a creature of the laws of the state in which it is created.
A corporation is formed by selecting a state in which to establish the corporation and to submit the required formation paperwork usually to an office known as the Secretary of State. The following link provides submission information for each state.
As a final note it is important to note that a corporation formed under the laws of a US state is subject to federal and state income tax on its worldwide income. The tax is levied on its net taxable income. Further, as you are likely already aware dividends received by the UK holding company from other UK companies or from overseas companies should benefit from an exemption from corporation tax, however this exemption is available in certain circumstances so as to avoid double taxation and is subject to specific reporting requirements under UK law
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Good luck with the business!