The former Labour prime minister warned the infighting was likely to continue as there were too many divisions inside the Tory Party.
Days of turmoil in Number 10 have seen the exit of Boris Johnson’s special advisor Dominic Cummings and fellow Vote Leave veteran Lee Cain as communications director.
The fallout continued this weekend amid further allegations of personal attacks on the Prime Minister’s fiancée.
Asked about the recent developments involving the MP’s senior aides, Brown told the BBC’s Andrew Marr Show: “You’ve got 27,000 new cases yesterday, 500 deaths, you’ve got a million young people looking for work, you’ve got people without savings planning for Christmas.
“Really we should not be seeing this factional fighting at the moment.”
“I see a trade deal coming pretty soon. The Government simply cannot afford to be at war with America on the one hand and Europe on the other at the start of the new year.”
Speaking from his home in Fife, Mr Brown added: “This is not simply a problem of a dysfunctional Downing Street, this is a dysfunctional UK, and we really need a wholesale, root and branch constitutional review to bring people together at a later stage.”
Environment Secretary George Eustice said Mr Cummings’ exit will have no impact on crucial Brexit negotiations on a future trade deal.
Speaking to Sky News’s Sophy Ridge On Sunday, Mr Eustice said: “The negotiations have been led by David Frost from the beginning. He’s got a very talented, experienced team of technical experts around him.
“He’s led these negotiations from the start and obviously remains in place and continues to do so.
“So, I don’t actually think the departure of Dominic Cummings makes any impact on the negotiations, since Lord Frost has been leading those.”
Mr Eustice said next week is “when things need to move” for the UK and EU to agree to a trade deal.
He said: “Both sides recognise that time is very, very short. It’s not long ago we were saying we needed to get some kind of conclusion by the middle of October.
“People have persevered with these talks. There does come a point frankly where businesses need to know what they are preparing for.”
The comments came after it appeared Johnson will attempt to reassert control over his Government by meeting with concerned Conservatives following the power struggle.
The Sunday Times reported the PM will “attempt to get his premiership back on track” by establishing a policy board that will appeal to northern working class voters who helped Johnson win last year’s general election.
The paper said the group will be chaired by MP Neil O’Brien, who helped former chancellor George Osborne devise the Northern Powerhouse, and added Johnson will meet the Northern Research Group of MPs on Monday to listen to their concerns.
Tory MPs complained the party and Parliament were not being heard during the time Mr Cummings held sway.
Former attorney general Dominic Grieve was scathing about Mr Cummings’ time at the centre of power.
He told Times Radio: “I consider that he’s created mayhem in government.
“The whole of his period in government has been marked by a slide in standards so that the Number 10 press office has been used as a vehicle for distributing smears, untruths and lies which was very obvious in the autumn of last year and the period around prorogation and the run up to the general election.
“This year it’s difficult not to say that he’s created nothing except chaos with the Prime Minister.
“Whether it’s the handling of Covid and his own behaviour, whether it’s the Internal Market Bill because quite apart from being utterly wrong in violating international law, that has blown up in the face of the Government and led to a massive rebellion in the Commons and the House of Lords and something of a crisis associated with that.”
According to reports, tensions were heightened in Downing Street when the PM was shown “hostile texts” briefing against Miss Symonds, which had been forwarded to her.
Theresa May’s ex-chief of staff, Lord Gavin Barwell, also said the departure could lead to more harmonious relations between the Johnson and his MPs.
Referring to the PM, Lord Barwell said: “It feels to me that there’s an opportunity here for him to get his Downing Street operation more harmonious and more effective.
“To rebuild relations with Conservative MPs, the parliamentary party.
“And, perhaps, to set a less confrontational and more unifying tone, that is maybe more in tune with his natural instincts.”
Sir Edward Lister was announced as the interim chief of staff pending a permanent appointment, with the Sunday Times reporting chief Brexit negotiator Lord Frost, Lord True and Lord Bridges of Headley are each being considered for the role.
The dramatic events have come as Brexit heads to a crucial phase next week, as London seeks a trade deal with Brussels before the end of the transition period on December 31.
The PM’s official spokesman James Slack insisted Johnson was not being distracted by the row.