it might go on if you are not crucial, you might be kept till the trial is over. AND possibly fined for each day they house you.

You might also be given the chance to explain why you answered "no", and if the reasons are valid (e.g., religious beliefs), you would be subjected to an alternate procedure to be "sworn in". I would think, though, that counsel would be aware of such objections beforehand and would make appropriate arrangements; it probably wouldn't arise at the moment of swearing in.

Well, it might also tick off the judge, which is very very rarely a good idea.

@Panzercrisis right.

In UK courts, witnesses are processed by court staff prior to giving evidence. Assessment for travelling expenses, verification of identity, etc. Also they are asked whether they wish to take a religious oath or else affirm that they will tell the truth. Courts keep holy books for the major religions. Once, a court clerk told me, a witness told her she needed to break a china plate or a glass so the court cafeteria supplied a saucer, wrapped in a clean dish towel, which was duly broken. Witnesses don't get to appear to give evidence until an appropriate declaration method is established.

Surely any properly run court in a jurisdiction with freedom of religion would be checking this before sending the witness to the 'stand' or whatever the local term is?

@Panzercrisis in the UK, there is an official wording for the "affirmation". "I swear by Almighty God that..." is replaced by "I do solemnly and sincerely and truly declare and affirm that..." There is also a "promise" form used by witnesses under the age of 14, who can not legally swear an oath: "I promise before Almighty God that ..." The various wordings are fixed by law. The words "Almighty God" can be replaced by the name of any recognized deity, e.g. Jehovah - but the Flying Spaghetti Monster probably isn't acceptable.

@Panzercrisis, united states - Can you refuse to swear on the Bible? - Law Stack Exchange answers the question of specific forms of the oath. ¶ I suspect that most courts would accept any declaration that indicates that you intend to tell the truth and that you are aware that deliberate errors or omissions would be subject to perjury charges. But this question is about when one simply refuses to make such a statement, regardless of its form.

If you claim that you are an anarchist and that is a religious belief, not a political conviction, (after all, ideologies and religions are pretty similar at a philosophical level) and your religious belief makes any book connected in any way to the state, "disgusting" for you - would that fly? (Possibly adapted slightly)

@d-b Check the linked question in the comment above yours: In the US you don't have to put your hand on any particular object (book or otherwise) if you don't want to. Seems like from previous comment it would be similar in the UK (given the breaking-a-saucer story).

It also just depends on the court. California usually uses "You do solemnly state that the testimony you may give..." and just raising your hand without the use of a religious book, which avoids most issues and averts the need for a "do you want to swear or affirm?" conversation. But if you answered no, I suspect the court would inquire what's going on and then decide how to proceed depending on your reaction.

@d-b - if you were so disengaged from the state, why turn up at the court?

There is a long history in the US of replacing 'swear' with 'affirm' to address religious or ethical issues with swearing, notably including the Presidential Oath of Office as spelled out in the Constitution.

@alephzero What forms and questionnaires do deities have to complete in order to be recognized?

@arp correct. There was a big debate in the Constitutional Convention about it.