I work in adult social care, here's what happens in my area.
Call from carer saying they've reached their limit, this will trigger an urgent call for assessment by the rapid care team, their aim is within the hour. They will assess exactly what care is required and any referrals to other services (health,telecare, pendant service etc) and will immediately create a care plan stating what is required and how this care looks. Some people will be taken into immediate respite (care home), this may or may not be funded by the patient. This is all dependant on care needs and they can usually be met at home.
If the care is sustainable by a standard care package, then a referral will be made to that area team, they will be assigned a social worker and the care will be put in asap by the local agency, in the meantime, the local authority will provide the care.
At no time at all would any member of the family or support network be judged, quite the opposite in fact. It is extremely common for carer breakdown to happen and is the direct reason for some hospital admissions for the carer as they have literally burnt themselves out.
DO NOT feel guilty, DO NOT feel that there is no support for you both. Ring social services and explain the situation, they are there to help.