Being mindful of the fact that different groups faced different circumstances, roughly how long did it take for Allied soldiers to get from the boats to the tops of the bluffs?
Well it entirely depends on which beach/landing zone you're referring to.
By bluffs, I'm assuming you're either referring to Pointe du Hoc or Omaha Beach (the beach seen at the beginning of Saving Private Ryan).
Pointe du Hoc was assaulted by members of the 2nd Ranger Battalion, who had to utilize rope ladders in order to scale the cliffs. Most accounts state that the Rangers were able to scale the cliffs and reach the top (and hold positions there) within 45 minutes or so of landing. However, one could argue this was the easiest part of their operation, as they were tasked with destroying two artillery emplacements (which had been moved farther from their landing zone prior to the landings), clearing a blockhouse that was being used as a potential spotting station, and to secure and hold the area against German counter-attacks for 2 days before being properly relieved.
Omaha beach (which was sub-divided into another six beaches and the beach most people think of when they think of D-Day) on the other hand, was arguably the most fortified and heavily contested beach. Initial landings started around 0620-0630, and due to the initial Allied naval and air bombardments having minimal effect on German positions, the first and second waves faced intense machine gun and artillery fire supported by a regiment sized force (about 5000 men give or take a few additional companies) entrenched along the beachhead's bluffs and cliffs. Difficulties with landing troops in their correct landing zones due to tides and weather added to the chaos, as many of the soldiers found their commanding officers/NCO's either absent or incapacitated upon landing. Another issue is that many of the troops found that they had anywhere from 100-300 feet of open beach (and sometimes they were unfortunate enough to have landed on a sand bar, often finding that they had to traverse through water again before reaching the actual beachhead) between themselves and the base of the bunkers/cliffs which obviously is a huge obstacle to clear when you're carrying as much as 30-50 pounds of gear and you're soaking wet to boot. Fortunately, enough NCO's and officers survived that they were able to gather groups of men in the Eastern most portion of Omaha and lead men off the beach to assault the entrenched positions at the top of the bluffs and begin a breakthrough of the German defenses that eventually led to the establishment of an Allied controlled beachhead (this was basically the plot we see in the opening of Saving Private Ryan).
Anyhow, to answer your question (in regards to Omaha beach), from the first wave landings at 0630 to American soldiers actually getting to the top of the bluffs and holding those positions, the Americans did not have an established presence (or at least once that could remain) on the bluffs until roughly two hours after the initial landings at 0630, so roughly between 0815 and 0830. By 0900 enough of an Allied presence existed at the top of the bluffs that the beachhead was initially secured and more reinforcements could be brought in as remnants of the first three waves made their way inland to repeal any German counter attacks.
As far as the other beaches are concerned, those are all vastly different experiences (and if I don't get to it later on as I'm about to go on an errand run, I'm sure someone else can) but are worth hearing about. To give you a general answer to your question, most beaches had an established beachhead by mid-day on June 6. By D-Day +3, Gold and Omaha beaches had their mulberry piers and beach masters running (temporary piers for offloading men and equipment) so that even more men, material and equipment could be offloaded to help with the push into France.
Sources:
D-Day: The Battle for Normandy, by Anthony Beevor