How long will it take for social security to bill for my Mom's nursing home care in California?

Sasha, I am not clear on what your question is. Is your Mom still living, or has she passed and you wonder how long it would take medicaid to file what they expect in recovery from her estate.
Or is your Mom living and you wonder how long it will take the Nursing Home to arrange for your Mom's Social Security to go to them?
I just am not certain exactly what you are asking us.

Social security isn’t paying for your moms nursing home so they won’t be sending a bill. Neither will medi-cal.

I'm not in CA but my MIL has been on Medicaid since 2017. There is usually a 3-month time period from when the app is submitted to the letter of acceptance. In MN the state deals directly with the facility in terms of billing and payment. We as PoA get a copy of the monthly rent but SS deposits her money directly into her checking account (we set this up) and then the facility takes it out directly from her checking and Medicaid takes care of all her medical expenses and we just get the notification paperwork. Not sure if CA takes longer because there's so many more people...?

Social security will not bill for nursing home care.

Are you meaning a letter from the state to recover expenses paid for her care under Medicaid? In your case MediCal since you are in California.