Do you think it's worth a shot asking immigration to hurry the process up so we don't have to pay for new visas?
I don't think you can really say 'hurry up', but you can ask them if they think it's possible that the Residence visas will be through in time, and also, if not, what they recommend as the best course of action for you, all things considered.
You should be eligible to get a partner-sponsored visa, to match your wife's graduate job search visa for length, and also a matching one for your elder child(ren), in case the Residence won't be through in time.
Good luck with it all, and yes, do keep us posted with what happens.