Ttykuu

すみません、 おそくなって。ずいぶんまちましたか。

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?

Most important point: Do not rely on Google Translate. It is no good a Japanese.

There is no information about tense in おそくなって。This is an incomplete sentence. But you can know the tense even without the next sentence by context. If someone calls you before you are due to meet they wouldn't say "sorry I was late", and if they are talking to you after they have met you they wouldn't say "sorry I will be late".

So why is おそくなって left as an incomplete sentence? As you said, the て-form is used to link two clauses. The second clause is just left unsaid. It is up to the listener to fill in the missing information. This is quite a common thing that you'll see over and over again. It is most likely to be some feeling of regret in this case.

In fact in this case it's probably nothing more than just おそくなってすみません (I'm sorry I was late) with the parts swapped round.

As for the difference between "I'm late" and "I'm running late", personally I can't see much of a distinction in English. I think you may be over-analysing it.