A Civil ceremony can, by law, only include music and readings that are not religious. On the other hand, your venue is much more likely to be suited to a string quartet, harpist or other quiet instruments and it will be possible to use recorded music - make sure that you have permission from the venue for this as there may be a fee payable to the performing rights society.
Any piece of instrumental music that is not specifically connected with a religious
setting can be chosen.
Examples of songs that are not religious could be:
As well as any songs from the musicals of the West End or Broadway etc.
Music For a Church Blessing - Brides Guide to Wedding Music - Orders Of Service and Other Useful Things