If you have to apply for help with your rent and you move to a different local authority then yes you will have to apply for UC to claim the housing element.
If you move to a different council/local authority as you own your own property then you do not need to claim UC at this moment in time, CB ESA and PIP are not affected when you change local authorities, you are only affected if you have to claim housing benefit.
Gary
Nothing on this board constitutes legal advice - always consult a professional about specific problems
