No, the batteries typically do not last a week. More like ~8-ish hours. But there are typically also backup generators.

The impression I have is that the batteries are, in part, sized to give you a chance to fix the generator if it doesn't immediately start up.

Sorry, I was speaking about the US POTS. My apologies for not being clear, especially when commenting on a post from a UK site.

CO buildings in the US still maintain that infrastructure even though the buildings themselves are full of broadband hardware and an actual CO is now just a block of RAM