Most of the world manages this without toilet paper, the trick is to be as hygienic about it as you can.
I lived and worked in India for two years and was often out of the city. I learned right away to go without TP and am glad I did: it freed me never to have to worry about it. Western style toilets were rare to non-existent in most of the places I went, including my own house.
There really several aspects to the process:
1) Learn to squat and be comfortable at it. Indian style toilets are low to the floor with foot rests to step and squat on to help you aim. We are a sitting/chair culture, not a squatting culture, but you can get used to it with practice just like anything else.
In the outdoors, try digging a cat-hole in a more rectangular shape so you can look down and see the front of it to help your aim.
2) Clean yourself always and only with some water and with one hand; your left hand is the preferred one in India, as most people are right handed. But be consistent.
3) Use water to clean that hand immediately while still squatting. Indians typically carry a small pot of water (known as a lota) for this. Houses have water taps at hand squatting level to do this.
Outdoors, put your water bottle at the ready as you squat. Pour from your right hand onto your left to clean. When done, clean your hands again.
4) Be aware of your hands. Clean them whenever practical and try to keep them clean. Indians are very aware of handedness and using their 'clean' hand for handling food, touching people, etc., around which there are numerous social norms.
I use these methods and clean my hand with biodegradable soap. I carry some Purell to clean with before I handle food or water.