Medicaid and joint account; What happens to the balance after Mom passes?

Being your the child, and not a spouse. I would've kept the accounts seperate, and named myself as the POD on her account. That way, it would bypass MERP upon her passing.
You didn't specify how much is in the joint, but if it's over $2000 that's asking for trouble with Medicaid, in many ways. For your question:
MERP will presume, half the money in that joint bank account belongs to your mom, and go after it.

Is your situation that the money in this account came from BOTH deposits of moms AND deposits of yours? So that the funds are actually commingled. Is that what’s happening?

If so that’s way way different that if it’s moms account, it’s only her SS# tied to the account, and your a signature on the account and maybe your also POD (pay on death) on the account. For these, all the $ there is all moms asset for Medicaid.

but if it’s commingled, you’ve got to establish whose assets went into the account and then show which checks written or withdrawals done that were purely hers and which were purely yours. So that it clearly shows wasn’t gifting of $ from her to you. It’s a forensics accountant project. Bigger CPA firms will have a forensic team on staff. The pit bull type of divorce attys will have freelance accountants they work with to do forensics. If you have years of bank statements and check registers and both your & moms old taxes & w-2, 1099 stuff, a good forensics freelancer can do a report easily. The report the accountant signed off on on their letterhead and you give it to Medicaid to establish why $$$$ in the joint acct is yours and no an asset of moms subject to a spend down.

if you are still commingling $, please please stop. Every month this continues can cause a gifting inquiry later.

Make sure moms account is POD to you. If so it will not be a probatable asset so passed to you in its entirety after mom dies and will not be a factor for MERP / estate recovery. The rub will be keeping it under the maximum allowed by her states Medicaid or her actually dying with any $ left. For most states (not NYS it seems) the max allowed in liquid assets is $ 2,000. That’s it, 2k. If you are hoping mom will die with $$$ left in the bank for your to inherit, if they are on Medicaid, is unlikely for that to happen.

I don't understand what you mean by you pay the spend down. To be on Medicaid Mom had to have met a cap, in my State its 2k. Thats all she would be allowed in assets. Her home and a car are exempt assets. The only thing you would need to pay is her SS and/or her pension to the NH to offset her care.

Medicaid looks at all accts with Moms name on it as hers unless u can prove differently. So hopefully only Moms money has gone into that account. Then its how its worded that determines if its yours. Jointly with survivorship? This is a question for her bank.

What will happen when Mom passes you will become her Executor if there is a Will or need to become the Administrator or sign an affidavit (estates under a certain amount only need an affidavit) to be able to get a short certificate if no Will. The certificate will give you the ability to handle whatever you need to do. Mom has a Personal Needs Acct (PNA) with the NH. Any money left in that acct is part of her estate. Same with any money in her banking account, unless you have survivorship. You will need to pay any bills she may owe out of her "estate". As an only child, what is left will be yours.

Be aware, that if where you are living is Moms house, Medicaid will try to recoup the money they spent on Moms care. If you were her Caregiver before her admission or were a resident when she went in, you maybe able to stay in the house but may have to prove you can afford to pay bills. A lean will be placed on it. If you leave, sell or die, the lean will need to be satisfied.