I don't think there is a problem with some programmes on the BBC being religious because presumably if there is a gap in the market for talking about atheism on TV the producers would commission a series to fill that gap.
I think there are two types of religious programming - the first being programmes about religion, with the aim of informing and educating the audience about some aspect of religious belief. I don't think such programming is controversial, provided is is done in an appropriately balanced manner.
The second type of programme is, in effect, broadcast worship of some form or other, specifically aimed at meeting the religious needs of religious people. Here is where greater care needs to be taken. While I have no great issue on principle care needs to be taken for a public service broadcaster such as the BBC, who needs to be impartial, from being seen to support or even promote particular religious views without balance or challenge. So if within the broadcast of a religious service a minister of religion is permitted to make a statement of faith and opinion (which they will often do within a sermon) which may go well beyond pure religious views, then in the form of balance the BBC needs to ensure that those views can be balanced by other views and challenged appropriately.
In the world of politics, while the BBC might broadcast a minister's speech at a party conference, it will ensure balance by also broadcasting speeches from other parties and having some editorial opinion/challenge from BBC journalists. This never happens with religion. And of course the most egregious example is Thought for the Day - a slot smack in the middle of BBC radios prefer news show, which allows a person to promulgate a specific view, uninterrupted, without balance and without challenge. And that person is not permitted to be someone who is not religious. It is the equivalent of having a 3 minute slot every day where a politician from one of a number of right wing parties can provide any opinion they wish, completely unchallenged, but a left wing politican is never allowed to give their view (or reverse the left/right). That simply isn't right.
Is there any evidence of an appetite amongst the British public for such programmes that has been vetoed by the BBC?
The BBC has been challenged repeatedly to open up Thought for the Day to non religious people who may have just as 'valid' moral and philosophical 'thoughts' pertaining to the issues of the day compared to religious people. The notion has been repeatedly vetoed by the BBC who continue to ban non religious people from the slot.