What effect does the house's repetition of the time have on the mood of the story?

In literary terms, "mood" is the emotional atmosphere evoked by a piece. In this story, the house's repetition of the time draws our attention to a devastating, sad change. It also contributes to the feeling that the house -- while managing to keep its operations under control -- is disoriented and abandoned. Let's look more closely at the details.
First, consider the context. The house is a mechanical servant to its human masters, but its masters are dead, killed in a nuclear holocaust. Thus, the time announcements underscore what is missing now -- the people who used to live by this schedule, and the wider civilization that made this schedule possible. You can observe this at various points in the story.
Sometimes, the voice-clock doesn't just tell the time. It also urges its potential listeners to do something:

"Eight-one, tick-tock, eight-one o'clock, off to school, off to work…"

On other occasions, the voice-clock states only the time of day. But nevertheless, the house performs specific services at these times.

"Five o'clock. The bath is filled with clear hot water."

Either way, the time messages remind us of the schedules these people followed, and the daily activities they engaged in. We recognize these schedules and activities, and so we feel empathy.
In addition, we are aware of the scale the loss. It isn't just one family that died. Very likely, civilization has been destroyed, in which case the schedule is no longer relevant to anyone. Thus, the time updates highlight this sense of irrelevance, and help contribute to an atmosphere of futility. Time is passing, but there is no one left for whom this passage has any meaning.
Finally, the time messages help emphasize the house's ignorance of what is going on. This contributes to the impression that the house is a confused or deranged -- and possibly lonely -- being. And it heightens the sense of sadness.