How your caregiver responds will be individual to them. While in recent years there has been an acceptance of additional education for caregivers about how to relate to and care for parents who have lost their baby, not all will have had this training and even if they have, their own personal views and perceptions of death can override 'what is expected'.
Their responses may also relate to their previous dealings with death either as a professional or on personal level. Each caregiver will always have 'a first', but previous experience may not necessarily be a help, or they may put their own experiences onto you, whether you want them or not.
You may find that your caregiver naturally responds with crying, or they may just try and 'hold things together' until they are alone. Some caregivers can seem quite indifferent and insensitive, perhaps they are wary that you might blame them for what has happened (whether they were responsible or not). Some caregivers will have a natural ability to do and say all the right things and be in tune with your needs, others may seem like they are more of a hindrance than help. If you feel you want to deal with someone else and this is possible, ask for this to be arranged.
Your caregiver may play the role of a counsellor to an extent or they may feel this is beyond their level of expertise and refer you onto someone else who specialises in this. They may help you make decisions about your involvement with your baby after they have died, assist in obtaining keepsakes for you to keep, take blood from the woman or make available information about any arrangements that need to be made. They may visit you at home if you go home early or you may be sent another carer who is able to fulfil this role.
It is not compulsory, but most caregivers will try and make some contact with women at some later time to see how they are or go over the details of what happened or just extend their sympathy and offer support.