What partition is needed to wash your hands in a bathroom for bread or davening?

Question

I heard rumours that after walking into a bathroom, nowadays, you're required to wash you're hands three times (as in, I've seen some Rabbi do this: after just walking into a bathroom to get something, he washed his hands 3 alternating times).

I also heard (and experienced) that if there's a partition between you and the actual bathroom, you can wash your hands there, even for bread and davening. I'm not sure though as to the extent of the partition required. In the bathroom at the shul where I go to, they have the netilas-yadayim cup in a sink that's inside the actual bathroom, with only a partition separating between it and the urinal, and the other toilet is right next to it (but that is fully partitioned off). Would something in that case still be considered a valid partition, even though it only blocks the urinal from one side, but you can still see it in the front, or does the partition have to block it from all directions, or is only a closed door sufficient?

Basically: what is the extent of the partition needed to wash your hands in a bathroom for bread or davening without any halachic concerns whatsoever?

Answer 1

First of all, it is not clear one cannot wash hands in a clean modern-day bathroom (see e.g., here). But since there are opinions which require washing outside a bathroom and you ask for an option "without any halachic concerns whatsoever", I found two relevant solutions

1. R Daniel Mann from Eretz Hemda (here, speaking of making blessing in the bathroom which is a higher level of requirements) writes that "if one can cover the toilet all around or get a 30 inch partition in front of it and smell does not emanate from it, this problem is solved" (based on SA Orach Chaim 76:1)
2. dinonline (here) mentions a "halachic partition" formed of 2 vertical wooden beams on opposite sides of the wall with a string joining them

Of course, consult your rabbi before implementing anything you learn here.