Housing benefit and ESA

I have just been transferred from IB to ESA support group and receive housing benefit. Do I have to tell the Housing office about the change myself or will they be informed anyway?

Is there likely to be any delay in rent payments because of this? MY housing benefit is paid on the 28th and i start ESA on the 25th so I am worried about the dates being so close together if there is a delay?

Thanks every so much.