I do OAOL pushups every day on top of my S&S training, but I do not do them at the same time of day. I think if you're doing S&S in a more serious way, you're doing it with full power and not taking too much rest betwen sets, so in my case I'm quite "done" after my S&S workout. However, at another time in the day, I do the warrior pushups. My S&S press is the TGU of course, and I'm pressing 40kg with it. I weigh 98kg so my warrior pushup is (apparently) 68kg on my one arm, according to my handy dandy bathroom scale. The warrior pushup is therefore for me a much heavier and harder press than the TGU. I don't do the pistols any longer because I don't see any practical use for them in my life - the swings and TGUs take care of my leg exercise as much as I need, and I can still do 75% pistols on demand anytime anywhere as a result of these. I think the big gap for bodyweight training is the lower back. I happen to be born with a weak lower back, which makes kettlebell swings my ideal workout focus. If my TGU was being done with the same amount of weight as my warrior pushup, I would not need to do the warrior pushup any longer. As things are, I don't want to give up my "warrior strength" by stopping my NW training entirely.
Anyhow, simply put, I've been doing warrior pushups and S&S together since the start of September and things have been very good!