I googled the sentence and found it's from a LISTENING comprehension test. I'm sure that a pause must be put after 降[ふ]って in the audio file, just like 雨が降って、来るのが遅くなったので. (The intonations are also different between 雨が降ってくるのが and 雨が降って、来るのが.) I, too, thought the subject of 遅い must be "rain falling", not "I", when I saw your sentence.
As for your initial question, yes, the の in 来るの is a nominalizer. (Purly grammatically, the の in 遅くなったので can be considered also a nominalizer.)
雨が降って、来るのが遅くなったので
since I was late to come (to the office) because of the rain