OMG, I got through this. I still can't believe it. This forum thread gave me what I needed though, but I'll toss in a few other things that I felt helped because I was ready to just forget about this darn thing.
1) Obviously practice that hard as heck 'special' jump. I just climbed back up the stairs to the previous room and practiced on the big door. Even after reading this forum it took me a while before I actually got "E" to jump sideways, and it was like, "Whoa, what did I just do?" Then I'd try it again and either get nothing or worse, the dreaded 'reverse' jump. I'd get about 5 practice jumps correct in a row and then I'd go 10+ with bad jumps. It just took a lot of practice to get those left and right sideways jumps to work out. Even when I finally got through this I was still only getting the jump right about 80% of the time.
2) Even if you run out of time no matter how much progress you made on that particular run just keep practicing that section. Just swim to the last location and go through that section over and over. Oddly for me the section I struggled with was jumping up those darn stone steps. One false move and you are down in the water. Each section has several places where you just need to stay calm when you 'aren't doing the special jump.
3) There are only three places where you need to make the special jump. The first wall, the second jump from the 'round' pillar, and then the third jump which gets you over to the lever and drops you down before you start heading up the stone steps. For some reason I thought that when you reached the wall where there is water dripping down that you needed to do another special jump. You don't. It works, but you don't need to do the fancy jump. Just turn side ways and jump up on to the ledge.
4) Sleep on it. For whatever reason whenever I've encountered a section that I can't get through on any game and I feel 'stuck' I just leave it alone for a day and come back the next. That's what I did on this one. I spent hours the first time trying to get through this but by then my nerves were too tight so I was making a lot of stupid mistakes. I tried it again this morning and within 45 minutes I got through.
5) Turn the brightness level all the way up. It was somewhat 'foggy' but it helped to see the darker stones, logs, etc.
Again though the biggest advice is that even when the clock runs out just keep going and practicing in that section. Each progressive has a place where you can swim back to and do that section over and over again. Then when you feel confident go back to the beginning and start the timer and go for it.
Total time spent to get through this - about 8 hours for me, and that was with constant practicing. Good luck, but you can do this.