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.
Some examples:
"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.
"To this day" is an unusual phrasing, only used to talk about something that has been going on a long time and is still that way. It would not be used for short time periods or indicating the turn-over of a calendar day in the minute past midnight. You should generally think of not using it for anything much less than a year; and usually much longer spans of time.
For instance, a difficult problem known to the ancient Greeks might turn out to be "still unsolved, to this day." And to put stronger and more mysterious emphasis on how odd it is that no one has figured it out, you might phrase it as "still unsolved, even to this very day".
Going for something less dramatic, the casual "until today" doesn't specify a clock time. If I said "I didn't know that until today", I am not suggesting I got an email about it at midnight. I am saying "I didn't know this fact prior to the current calendar day, and only found out at a clock time prior to us speaking now."
(Note: We sometimes consider a very late hour like 1:00 AM to be the previous day's "night". Hence I might say "I saw them yesterday night" on Monday, even if I had actually seen them earlier on that same Monday at 1:00 AM.)
To get more specific about "day" you have to modify it. There are phrases like "You will receive it by the next business day" which means that if someone says that on a Friday, you will not get it until Monday (assuming Monday is not a work holiday). "The next day, she opened the letter" or similar phrases are ambiguous about time.
You will have to be specific if a specific time is meant. For instance: "As the clock struck midnight, the phantom appeared."
That is all very far from the understanding when people would use "to this day". Once again, it is only a rare phrasing used to indicate a long time frame extending all the way up to the present...usually for dramatic and/or mysterious effect.
Best Answer
Here prior is being used as a shorthand of sorts for a fuller phrase such as "prior to meeting him here" or "prior to coming into the room". It is technically not correct usage, but it is probably relatively common in some areas. (Since I don't know where "Bazan", the speaker, is from, other than 'not from my area', I can't really say how common it might be; but it's not common in my area, at least.)
"Going about his day", as noted in the comments, is an idiom meaning to do all the normal, run-of-the-mill, routine things that you usually do every day; shower & shave, eat breakfast, read the papers, check your e-mail, get ready for work, and so forth.