Hello.
As soon as you get the formal offer of a skilled job, you can take/send the proof to the LOCAL INZ office (not the one overseas that handled your original application), and they will activate your deferred Residence straight away. That is, you and any family members will get the full Residence visa labels put in your passports. Residence issued in that situation has a Section 49(1) condition upon it, that you, the main applicant, must work for at least three months in the job you have told INZ about - after that time, you can go back to INZ with evidence (pay slips, tax papers) that you have done so, and are still with that company, and they will, without further charge, give you another Residence label with no conditions on it. After that, if you want to, you are free as far as INZ is concerned to change jobs (details depending on your contract, of course).
Otherwise, if you wanted, you could work for three months in the new skilled job, and THEN go to INZ with your proof, in which case you would get your Residence visa without any conditions on it straight away.