I thought the meaning was :
Sorry I am running late. Did you wait long?
But when I googled it, the おそくなって means "I'm late." Is this one of the usual phrases used in Japan like おなかがいっぱい or のどがかわいた? Because I could not understand why it is I'm late rather than I'm running late considering て form of なる is used unless て here is used to link sentences which I doubt because there is a 。before ずいぶん.
So why is it なって? Why not なった? Do we always need to check the next sentence (because that one is in past tense)? Is て form used to link sentence?