Which word(s) can be used instead of "the first day of the week?"
It depends on what you mean by "week" in this case.
In America, Sunday is considered the first day of the week, and Monday is considered the first day of the work week.
Note that SE sites end their week at the end of Saturday, GMT. This makes Sunday the first day of the SE week.
If you want to be safe you could just say "the start of the week," but you would still have to define what you meant by "week": the start of which week?
Rather than an alternative phrase, make it explicit:
... weekly, starting each Monday ...
or
... every week (Sunday through Saturday) ...