Hi sam
UC will inform both your local authority and HMRC that you have applied for UC.
You were told correctly, it is true that your HB should run on for a further two weeks when you apply for UC. I say should as it has been known that claimants have missed out, so you need to keep an eye on your account to make sure you get the run on.
Gary
Nothing on this board constitutes legal advice - always consult a professional about specific problems