Housing benefits are not paid further back in time then the month HMS receives the application.
It is therefore important to apply as soon as you can, even if your lease is not ready.
The first payment date is the first day of the month after the approval of the application.

Example:
You apply in January and the lease takes effect on January 1st. You will receive benefits on February 1st for the month of January.
You apply in January and the lease takes effect on December 1st. You will receive benefits on February 1st, but only for the month of January.