You can only ask if INZ will accept your application without the required birth certificate, giving copies of everything you have tried in order to get the certificate in time, but it's rare that that is agreed.
Here's a post on an old thread dealing with the specific problems some Indian applicants have.
http://www.enz.org/forum/showthread....183#post456183
For processing times, Google "NZ immigration (whatever branch) processing times", and you'll find the most recent estimate, based on how things have been going lately on average, but of course, an average won't necessarily apply to your specific case. These timings are never a promise, and have to be understood to be elastic.