Most likely the late 52nd-61st centuries. That's where AHistory puts it, too, and they're pretty good at nailing these things down.
But it's a bit weird. I have actually been curious about that era, and the order that we see it in. If I am remembering right, these are all the scenes from that period:
The Time of Angels / Flesh and Stone: This one is especially odd, because it is actually said to be set in the 51st century. River is attempting to earn a pardon, and she was released into Octavian's custody four days prior. A neat trivial bit is that we also see the Doctor and Amy visit the Delirium Archive in the 171st century, and it is said by the Doctor to be "the final resting place of the Headless Monks." Octavian's men are soldiers of the Church, but they do not seem to hate the Doctor, so either this is before the Siege, or they are not member's of Kovarian's renegade sect. For the sake of simplicity, I presume the former.
The Pandorica Opens: One of the only two 52nd century scenes with a hard date. 5145. But for River, this occurs *before* the Angel two-parter.
The Impossible Astronaut / Day of the Moon: River breaks out upon receiving the envelope (and she has her vortex manipulator, so this is after she blackmailed it off of Dorium in The Pandorica Opens), and then later returns and kisses the Doctor for his first time.
A Good Man Goes to War: The bulk of the episode is set here. We see the Silence and the origins of their astronaut suit plan. Kovarian says that River is "hope in this endless, bitter war against [the Doctor]." Colonel Manton is authorized to lower the hoods of the Headless Monks by "divine grant of the Papal Mainframe herself," which we now know to be the Church's flagship. However, this branch may have had their own Papal Mainframe ship, or, being religious extremists, they may simply be claiming that their actions are justified by their faith without them actually being so. Dorium closes down his bar and gets his head chopped off, so this is set after 5145 (where we saw him in The Pandorica Opens).
Let's Kill Hitler: The final scene, with River joining the Luna University, is set in 5123. However, the Teselecta ship that appears in this story is from the 52nd century era, and provides the Doctor with information about the Silence. However, it does not know what the First Question is, which could suggest that the Siege of Trenzalore has not yet begun from its perspective.
Closing Time: Some years after Let's Kill Hitler's closing scene. River has earned her degree, but is taken by Kovarian and forced into the astronaut suit to be left in Lake Silencio.
The Wedding of River Song: The Doctor hangs about in the 52nd century; tracks down and interrogates the Teselecta, plays Live Chess with Mark Gatiss the Space Viking, and then goes to visit Dorium's head in the Headless Monks' vaults, which puts this after the battle at Demons Run. Notably, Dorium - unlike the Teselecta - knows what the First Question is, but he had direct dealings with the Headless Monks and the Kovarian Chapter.
And then there is The Time of the Doctor (as well as The Name of the Doctor, but that can be set anytime after the Siege has concluded in the timeline where the Doctor died). Tasha Lem is the leader the Church of the Papal Mainframe (not yet renamed to the Church of the Silence). The Doctor decodes the First Question, which is what starts the Siege. Kovarian does not break away until 300 years into the Siege. The Trenzaloreans are human colonists, and are presumably of a certain Amish-or-Luddite-esque persuasion, judging by their technology level.
So, in what order do these events occur? Obviously, there is a clear line of progression from Pandorica -> Good Man -> Wedding. The latter two occur in or after 5145. But those are the only certainties. What about River Song going with Octavian to investigate the Byzantium in the 51st century, despite her younger self being imprisoned in 5145? My assumption is, this was a time-active mission - she was released into Octavian's custody in 5145+, and they then traveled back into the 51st century to when the ship crashed.
I also apply this logic to the Kovarian Chapter's actions. While one *could* try to say that Demon's Run was concurrent with the Siege of Trenzalore (nearly 300 years in) and that Kovarian's branch had only moved in space to get there, this would call into question why the Teselecta does not know what the First Question is, despite coming from that era. So, easier for me to just think that the Siege began in the late 52nd century; let's say, 5185, and that sometime before 5485 (when Clara first returns to Trenzalore, after Kovarian has broken away), the Kovarian Chapter travelled back to the early 52nd century (sometime after 5145) and carried out the Demons Run shenanigans. This explains why the Teselecta didn't know the Question (it was before 5185, when the Question got decoded), whereas Dorium could have simply learned what it was from the Kovarian Chapter, since they hailed from the future where the Siege had begun.