At the risk of TMI ( Is that possible in this thread?)...
Following on from colorectal surgery, I found that I had issues like this.
A need would come on quickly and un-expectedly, and really could not wait.
This did lead to rather more bears in the woods incidents than I would have liked.
Initially, I had no option but vegetation and water from my bottle.
Once you get over the initial yuck, the Arabic water and hand method actually works very well, and cleans your undercarriage better than a smear with dry paper.
I then took to carrying wipes, but it's settled down so I don't need to any more.
But I remain converted to the use of water to properly clean myself; I no longer feel clean with just a dry wipe from bogroll. Particularly in bike kit.