In an article in the Radio Times in January 2011, Webb revealed that his natural father was Peter Woods who was formerly a reporter with the Daily Mirror and later became a BBC newsreader.

He then worked as a reporter for BBC Radio 4's Today programme, before becoming a foreign affairs correspondent based in London and covering news around the world.

In 2006, at a seminar on impartiality, Webb said the BBC was anti-American and treated the US with "scorn and derision", according it "no moral weight".

[citation needed] In August 2009, Webb returned to the UK to replace Edward Stourton on BBC Radio 4's early morning news programme Today.

[13] Webb promoted that the era of the "big beast" news anchor would likely be drawing to a close, with the "very well-paid" John Humphrys and Huw Edwards "in the firing line".

"[15][16][17] In September 2021, Webb asked Liberal Democrat leader Ed Davey, if there should "not be spaces where biological males cannot go?"

[18] In February 2022, the BBC said that Webb was not sufficiently accurate when he described the philosophy professor Kathleen Stock – who resigned following protests over her views on gender identity and transgender rights – as being "falsely" accused of transphobia.

When living in the United States, their son Sam fell ill and was diagnosed with type 1 diabetes mellitus.

In 2012, Webb joined son Sam at Addenbrooke's Hospital in Cambridge, meeting Queen Camilla (then Duchess of Cornwall) as she visited the facilities.