It would depend on context. Typically though, at the very least the phrase is referring to 'late in the night on a Wednesday'. In general,
Late Wednesday night without any additional qualifiers could refer to any number of Wednesdays, though. It could refer to last Wednesday (the most recent Wednesday in the past), late at night. It could refer to the next upcoming Wednesday, late at night. It could also refer to any Wednesday in the past or future, at late at night on that day.
"How was your week?" asked Sam. "It was good. Though late Wednesday night John and I went to a party; just to find that it had been cancelled..." replied Jake.
In this case, the speaker is referring to the most recent Wednesday. Context is provided by the question posed by Sam. Even without that question, Jake's response uses the past tense, implying that a past Wednesday is being referred to.
"Three years ago, on a late Wednesday night, there was a murder in this apartment."
In this case, the speaker is referring to a Wednesday night from three years ago.
"In three weeks, late Wednesday night, we are having a party at my house."
In this case, the speaker is referring to a Wednesday night three weeks from now.
In all cases, the only thing that changes is which Wednesday is being referred to. The 'late' and 'night' only refer to what time of the day is being discussed rather than which day it is.