What is the ruling concerning the 'aqiqah for the first son if circumstances did not permit it at the time, and then the father was blessed with a second son and wishes to perform the 'aqiqah for him?

You are not obliged to perform the aqiqah for your son for whom you were unable to do so, because aqiqah is a confirmed Sunnah and not obligatory. If you are able to perform it for him before he reaches puberty, then you should do so. However, if he has reached puberty, then the obligation of aqiqah for him is lifted from you. It is not permissible for two children to share one aqiqah.