For Mr. Mandelson, the trick will be to keep Britain out of Mr. Trump’s line of fire — and to do it at a time when Prime Minister Keir Starmer’s left-of-center government is trying to reset its relationship with the E.U., to which Mr. Trump has long been hostile.

Mr. Mandelson has already been denounced by a faction of Trump loyalists, who failed to scuttle his nomination but may have played into his decision to apologize on Fox News last week for deriding Mr. Trump during his first term as a “white nationalist” and a “danger to the world.”

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