I wonder how does Waterway skip in this run work.

You can technically use other combo like

Apollo + Cockatrice Apollo + Unicorn

Basically with these combos we are suspending Nathan in air and when the summon ends, we are able to move him ever so slightly before re-activating the combo. Since the wall is too high to just wall kick + double jump over. We abuse the ability to keep are height by using the summon to get closer to the wall and kick off it again (in a normal run you are looking at 2-3 kicks off the wall) to get close enough to the top and be able to jump on top of the wall.

Are there any similar combos that don't require Apollos though?

Not as far as I know.

Have you made the tutorial yet, or does a tutorial exist? I've seen it used in speedruns but I haven't found any guides on how to do it.

Never mind, Garnitrex posted a tutorial.