Downing Street said that the prime minister would say in a podcast which has yet to be posted on the Number 10 website: "Never again should we face the deadlock that threatened to pull down these talks.
"Never again should we let a global deal to move towards a greener future be held to ransom by only a handful of countries."
He said lessons must be learned from the "tough negotiations" that took place in Copenhagen.
Ahead of the podcast - which No 10 had suggested would be published on Monday - Mr Brown used a video link from his Kirkcaldy constituency to tell a gathering of UK Copenhagen campaigners that he and Mr Miliband wanted the UN process reformed.
He said: "The United Nations needs to be in a position where we can get agreements with government working together without having these last minute negotiations where threats and fear can actually dominate the proceedings."
He added: "If America and China were able to show they were doing more and I believe that they can, then all countries - Australia, Brazil, Japan, Korea - all those countries that have ranges [of emission cuts] would be prepared to go to their highest level of ambition."
Mr Miliband said the vast majority of countries wanted a legally-binding treaty to protect the planet and he said that it appeared four or five countries were keen to "shelve the accord".
He wrote of a legally-binding treaty: "Some leading developing countries currently refuse to countenance this.
"That is why we did not secure an agreement that the political accord struck in Copenhagen should lead to a legally binding outcome.
"We did not get an agreement on 50% reductions in global emissions by 2050 or on 80% reductions by developed countries.
"Both were vetoed by China, despite the support of a coalition of developed and the vast majority of developing countries."
The accord was reached between the US, China, India, Brazil and South Africa, but is not legally binding.
China's Foreign Minister, Yang Jiechi, praised the summit in a statement which said: "Developing and developed countries are very different in their historical emissions responsibilities and current emissions levels, and in their basic national characteristics and development stages.
"Therefore, they should shoulder different responsibilities and obligations in fighting climate change."
"The Copenhagen conference is not a destination but a new beginning," he added.
United Nations Secretary General Ban Ki-moon says the agreement must be made legally binding next year.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.