My autistic disabled brother lives with me and just got prostate cancer. He has Medicare but we are trying to get Medicaid.

Your profile indicates that he is in a facility but your questions states he lives with you.
The facility should have a Social Worker that can help with Medicaid.
You could also contact a SHIP counselor in your area. They are trained to go over all Medicare and pretty sure Medicaid and figure out what is best plan for the individual person.
Contacting the Social Security office should garner some help with Medicaid as well.
But I think the most help would come from the facilities Social Worker.

Is Medicaid needed to remain in the NH? The staff should work with you on that; we learned that if the application is in, the patient's account can be marked 'Medicaid Pending' and Medicaid will retroactively pay for the months it takes for the application to be approved. The social worker at your brother's facility should also be helping.

And Medicare has additional coverage for cancer. That being said, prostate cancer is known to progress very slowly; hopefully that's the case for your brother. If so, cancer-related care may not be needed for years. See Medicare's publication at https://www.medicare.go/Pubs/pdf/11931-Cancer-Treatment-Services.pdf.

If you're part of an autism support network, there may be someone there who could share from their experience...

Yes they doctor can refer a social worker to help out with the forms .