As noted, you generally apply for a TIN, which can be applied for even if you aren’t a US citizen.
A taxpayer identification number (TIN) is a tax processing number required by the IRS (the U.S. tax authority) for all U.S. tax forms. Non-U.S. citizens may need an individual TIN (ITIN). If you are claiming a tax treaty benefit, you are required to provide either a Foreign TIN or a U.S. TIN. Learn more about TINs from the IRS.
HOWEVER, if you read the link Peggy left, you would see:
United Kingdom: Unique Taxpayer Reference (UTR), National Insurance Number (NINO).
So you should be able to use that as your TIN.
Please reference your local tax authority or seek professional tax advice to determine acceptable Tax Identification Numbers. Google cannot provide tax advice.