John, the OP already has an NIE, and his family as you say, will get theirs when they apply for residency (i.e. register on the EU citizens register - although most gestors probably still call this residency, so for the sake of clarity, let's for once just agree that it means the same thing!)
Although as you point out, since they are "moving" to Spain they should register immediately, in reality there's no great urgency to do so - although with school age children it will no doubt be in his interest to do so fairly quickly.
I would suggest that 1) he makes sure he has EHICs to cover the whole family for emergencies, and 2) get some quotes for private cover, as initially this may be simpler than figuring out his wife's employment status (which is probably a subject for another thread)
Having never used private health insurance in Spain, unfortunately I cannot recommend any providers, but surely someone here can answer his question? Sanitas are one of the best known providers, so maybe start there? They probably offer a family package.
http://www.sanitasexpat.com