If you are in the UK you can use the "accessible" toilet for disabled people. They follow a standard layout so can be easy to navigate. Increasingly often they use dark blue fittings to contrast with white walls and porcelain. (EDIT these are the $1)

Disabled toilets in the UK will often have a note on the door to remind passers by that "not all disabilities are visible." Ironic if you can't see to read the notice but still helpful if you are worried about public disapproval for not rocking up on a wheelchair like "proper disabled" people should.

When I enter a larger public bathroom like at an airport or something I will pull off to the side out of the way after I enter the door. Then I will listen for a few seconds, which is usually enough time to hear the general direction of the sinks, urinals, and stalls. Then I will use my cane and head in the direction of what I am looking for. Hand dryers are death though for trying to hear where things are located. Sometimes I will wait for the hand dryer to end before I start moving, but other times you just got to get moving and rely on O and M skills and hope that people will notice the cane.

If there isn't a single/family bathroom available, it comes down to listening and intuition. When I go into a public restroom for the first time, if I'm by myself I tend to just explore with my cane, heading towards the walls to see what I find. If I get to the sinks, I'll use that as a starting point, then logically work my way around the room, searching for stalls or urinals. Soap dispensers are usually either on the sink or installed fairly close to the wall, so a little bit of explorative flailing will usually find it, and then dryers or hand towels are usually at the end of the vanity, so I'll explore towards the walls and closer to the restroom exit to try to find the drying options. Will make large sweeps to find the trash if it's not immediately under the towel dispenser.

If there are others in the restroom, I'll listen and generally wait and ask for directions when someone is washing their hands. It can get a little awkward tapping someone with my cane who is standing at a urinal, but I just apologize and move on. It's super annoying when people go into stalls and use them without locking the door; I've walked straight into the backs of several people who do this and I tend to tell them off. As with anything, just takes practice to navigate around and confidence in your ONM skills to figure out restroom layout.

Either ask somebody there for help or feel your way around, go do your thing and wash your hands like you've never done before when you're done.

Yeah, I mean either listening or asking questions or just knowing. I mean most bathrooms are not a mystery.