Can a blockchain transaction history be updated by text messages using a mobile network? Is it possible to update Bitcoin transactions over mobile networks using texts or voice data transmission?
No, at present this is not possible. The Bitcoin network runs only over the internet. (Of course, there are lots of mobile data networks in the world, and Bitcoin works just fine over those.)

One could imagine some kind of service to receive transactions by phone or text and relay them to the Bitcoin network (and perhaps vice versa), but I don't think such a service currently exists.  I'm not sure that very many people would find it useful.

What applications do you have in mind for such a feature?