You should probably help your mother apply for Medicaid in New Jersey immediately, then apply again in New York as soon as she actually moves in with you. This means
two different application processes, but it is probably worth doing, for several reasons. First of all, her Medicaid coverage in New Jersey can be retroactive to the date she first applies (assuming that she is eventually found eligible), even if it takes some time to complete the application and have Medicaid make its eligibility decision. So, if you apply immediately and your mother remains in New Jersey for any length of time, Medicaid can help cover her medical bills in New Jersey from the date you apply until she leaves the state. Since it may take some time to relocate your mother to New York, this may save a considerable amount in medical bills. This makes particular sense because, as you say, her "future is unclear".
Likewise, you can help her apply for Medicaid in New York as soon as she moves in with you. If the Medicaid program in New York finds that she's eligible for coverage, that coverage will be retroactive to the date she applies in New York.
Also, once you have collected the various financial and other documents needed to help your mother apply for Medicaid in New Jersey, most of your work will already be done for applying in New York. You'll have to establish her residence in New York, but otherwise the paperwork for applying will be very much the same as it was in New Jersey. So, even though you'll be applying in two different states, you won't have to do everything twice.