Ms Bradley, MP for Staffordshire Moorlands, was formerly in the Department of Culture, Media and Sport and is considered a close ally of Prime Minister Theresa May.
Brokenshire has a small lesion on his right lung and will have surgery in the coming weeks.
Last January, Northern Ireland's government collapsed after the Irish nationalist party Sinn Fein withdrew from talks with the pro-British Democratic Unionist Party over renewable energy issues.
The MP for Old Bexley and Sidcup was a long-term colleague of Prime Minister Theresa May's, having previously worked with her at the Home Office.
Born in Newcastle-under-Lyme in 1970, Bradley moved to Buxton in Derbyshire when her family took over a hotel there and she went to the local comprehensive school.
Amid the chaos and confusion that attended this week's reshuffle, the announcement of a new secretary of state for Northern Ireland passed off with little notice.
Mr Coveney phoned Mr Brokenshire while on a visit to Cairo to wish him well.
In that time, there have been a round of elections in Northern Ireland (and the UK), Northern Ireland deputy first minister Martin McGuinness has died, and there have been many rounds of negotiations trying to restore power, but none have been successful.
He said: "His unwavering commitment - in public and in private - over the past year to securing the effective operation of the devolved powersharing institutions in Belfast has been hugely important".
While DUP Leader Arlene has said Northern Ireland will end up under Westminster rule if Sinn Féin do not remove their red lines.
"This is clearly a hard time for not only James but for Cathy, his wife and his children".
While the circumstances of his leaving are unfortunate, Mr Brokenshire's time in the north will be seen as well meaning but ultimately ineffective in terms of securing the restoration of the executive.
"James leaves the role with a very intimate knowledge of Northern Ireland and I look forward to working with him again in the future".
A relative unknown in Northern Ireland, Mr Brokenshire stepped into the spotlight, calling for snap elections for the Northern Ireland Assembly, early previous year following the resignation of Sinn Fein's Martin McGuinness.