In A Fire Upon the Deep, does anybody ever mention the date or at least the year that the book is set in?
Here's the introduction of the ship that retrieved Pham Nuwen's ship from A Fire Upon the Deep:
A dredge. The ship is about thirty thousand years old. Most of that time was spent in a deep penetration of the Slow Zone, plus ten thousand years in the Unthinking Depths.
So all we know is that Pham's ship had floated for an undisclosed length of time, and then spent 30,000 years traveling in the bottom-lugger. Then, he spent "a bit more than a century" sitting in storage at Relay.
The events in A Deepness in the Sky are impossible to pinpoint relative to Earth, but it does make reference to the Unix operating system and our first arrival on Earth's moon:
Take the Traders' method of timekeeping. The frame corrections were incredibly complex - and down at the very bottom of it was a little program that ran a counter. Second by second, the Queng Ho counted from the instant that a human had first set foot on Old Earth's moon. But if you looked at it still more closely ... the starting instant was actually about fifteen million seconds later, the 0-second of one of Humankind's first computer operating systems.
So all we have is a minimum: more than 30100 years after Pham Nuwen set out at the end of A Deepness in the Sky, which occurred many thousands of years after we first set foot on the Earth's moon. While the exact date isn't given, A Fire Upon the Deep gives a rough minimum (assuming that this memory of his is genuine):
When Pham Nuwen last knew [Qeng Ho], they had settled thirty worlds and were almost three thousand years old. (…) And of course, until Pham Nuwen was revived, no one in the Beyond had ever heard of it.
In the introduction to the special edition, the author notes that he used some dates for his own timekeeping, but these dates are not meant to be part of the story and are not definitive.
There is another style of date you will notice in some places, dates like 5 June 15989! This was my crude way of keeping track of the story chronology. Since Tines’ World days are about the same length as Earth’s, it was possible for me to use the calendar tools in Lotus Agenda™ to manage the chronology of the story. I more or less arbitrarily set the beginning of the story (the birth of the Straumli Blight) at 00:00:00 23 June 15988. Note that this assumption never leaks out of the annotations; it was just an artifice to make it easier for me to track the relative position of events occurring within the story. (In fact, it’s not certain just how far in the future this story happens; fourteen thousand years after 1988 seems to be about the minimum.)
This dating is in fact contradictory with the story (at least in appearance, since some of the records may have been faked by Old One), since Gondr states that
“The ship is about thirty thousand years old. Most of that time was spent in a deep penetration of the Slow Zone, plus ten thousand years in the Unthinking Depths.”
We know that Nyjora, where all humanity in the Beyond is descended from, was settled more than 3000 years before the story:
The mural was a famous one; the original had been an analog work from three thousand years ago. It showed life at an even further remove, during the Dark Ages on Nyjora.
In Deepness In The Sky they mention code archaeologists seeing references to unix epoch time. Even if they gave a date via that, we don't know how long Pham Nuwen was frozen down in the Unthinking Depths, but I would speculate that it must have been millions of years. And that's on top of the several (tens of) thousand years Deepness is set in our future.