The inventor of the world wide web, Sir Tim Berners-Lee, has won a partial victory against the BBC by getting it to let an atheist present an alternative Thought for the Day on its flagship radio news programme – but not at the usual time.

As guest editor of Radio 4's Today programme, Berners-Lee, an active member of the Unitarian church, wanted an atheist to present the three-minute daily religious slot, broadcast at 7.45am.

This was overruled, but the minister Andrew Pakula, who describes himself as a non-theist, was allowed to present an alternative Boxing Day Thought for the Day an hour earlier.

Berners-Lee said that as guest editor he felt he had a duty to poke at the BBC's traditions. "It was worth trying to point out that somebody who doesn't believe in God can still think," he said when interviewed on the programme about the experience of guest editing.

He noted that, according to Wikipedia, Britain is roughly split 50/50 between believers and non-believers. He also complained of a "certain amount of discrimination" against atheists, and suggested there should be a regular "Thought for the Day for atheists as well as a Thought for the Day from someone who believes in some supreme being".

In his alternative Thought, Pakula explained how he celebrated Christmas as an atheist minister of a Unitarian church in north London. "To me there is no inconsistency in being an atheist and celebrating Christmas," he said.

"While I don't literally believe the stories underlying Christmas, I do very much believe in its most important messages. Christmas reminds us that hope can come at the darkest times. It reminds us of the sacredness and innocence and possibility of children – that any child, however humble their circumstances, could change the world for the better."

Afterwards, Pakula questioned the BBC's approach to religion: "The BBC talks about not allowing people of no faith to present Thought for the Day. Well, what does 'no faith' mean? Here I am, a minister of religion, leading a congregation, talking about peace and love, and I'm considered a person of no faith because I say I'm an atheist."

Writing on Twitter, he said the BBC rules were arbitrary, as they would allow atheist Buddhists to present Thought for the Day.

"I'd say I have faith although I'm an atheist. BBC says this is impossible – except for Buddhists," he tweeted.

In August 2002 the prominent biologist Richard Dawkins, who went on to write the best-selling God Delusion, presented a secular Thought for the Day, but this too was broadcast at a different time.

On the Boxing Day programme the actual Thought for the Day was delivered by Jim Corrigall, a Unitarian minister in Ipswich who does believe in God.

Some of the Today presenters are known to be sympathetic to relaxing the programme's rules on religion. Last year Evan Davis, himself an atheist, told the Independent that Thought for the Day should be opened up to "spiritually minded secularists".

On the Boxing Day programme, presenter Justin Webb quipped that the world had not stopped spinning after Pakula's broadcast.