Hello Maverick Spirit,
We do not deal with Housing benefit queries on this site. With regards to how long the migration process to UC will take, this is virtually impossible to say. As you probably know the IB to ESA process was supposed to be completed in 2018 but this has now been extended to at least 2019. UC has already had significant issues which may impact on the timescales. Im sorry there isnt a more definitive answer.
Nothing on this board constitutes legal advice - always consult a professional about specific problems