If you dont like their choice of nursing home, state in writing why, they are then duty bound to offer an alternative, but --- they can find a worse one to force your hand into their first choice.If you can get/ask for a list of homes and visit each one, ask if they have vacancies and look around and then go back to the authority and say I have found this place, there is a vacancy and this is where we would like our relative to be placed, once they realise you are not going to be pushed around and you have done your homework they become more co-operative. Usually it boils down to either money they will have to spend or they are so overworked that they havent time to look around themselves and take the easiest/cheapest option.
Dont fall for " lets place them there for now " -- once there the relative will never be moved on.The more the place they are in now wants the bed the more bargaining power you have. Dont be afraid to speak out and visit the home they suggested at odd hours to get the complete picture and help your case.
Good luck, I have been there.