AFAIK, there is no restriction on the airlines that you can join if you're at the top end of the age scale. All cadets can be placed with any airline; who you end up with is dependent on the requirements of the various partner airlines when you come to the type rating stage of your training.
Regarding delays in training; it's impossible to say how soon after arriving in New Zealand you'll start flying. It's dependent on weather, instructor and aircraft availability etc. The first few weeks are very busy; in addition to induction you have the six NZ PPL exams to study for, as well as a type rating exam (Essential Knowledge Quiz) to sit for the aircraft that you will be flying. Therefore, you can expect your flying to pick up after this initial period.