I am not sure of the law either but an integral part of being a councillor is to represent the electorate and by logical extension, they must be advised of who he is and how to contact him. I too would be unhappy if a member ofthe council of which I am Chairman wished this. Quite why anyone would wish to be a councillor on the one hand but anonymous on the other is not clear to me.
The legal dept at your principal council or NALC would know, I am sure.