I am a NY lawyer. If you have been sent a waiver and consent, you are likely a distributee (next of kin) of the decedent. If you do not sign the waiver and consent, you will be served with a Citation. The citation will state a date to appear in court if you are contesting/ filing objections to the will. If you do not show up (or have a lawyer show up representing you) or you do not ask for an adjournment, the Court will likely enter an order admitting the will to probate. Of course, someone other than you may file objections and this would delay the probate. However, once the will is admitted to probate and the executor is appointed, he or she will marshall assets and distribute the assets pursuant to the will which he or she offered for probate. There is no specific time by which the citation will be issued and varies from county to county. However, if you are even thinking of contesting the will, you should consult a lawyer. Your post raises many issues and I have not gone through the issues in my response. If you want to discuss any further issues, this would be best done in a consultation.
Ms. Siegel provides sound advice - if you are considering contesting, you should consult with an attorney regarding the Waiver as well as how to proceed. There are only a handful of reasons to contest a Will's validity and the attorney will able to guide you in that repect