When we did our 4 weeks on the road (mostly bush camping, station and then Mum and Dad's before heading back) we did I think 2 nights at a Caravan Park - I threw all the nappies in while everyone else was cooking dinner (which meant the machines were mostly empty), then we hung them inside the 'front room' of the tent - this particular caravan park you weren't allowed to peg anywhere but the communal line ie so not in sight of others. Worked fine and they were all mostly dry by morning.
I had almost all PeaPods too with bamboo and hemp inserts, and used either a squirty drink bottle in place of our Little Squirt (holding over a toilet or bush loo), or if there was a flushing toilet (not always lol) then I used th flush to remove most of the poo. Otherwise toilet paper and then into the toilet.
The pee'd ones, I squished with my feet in the shower for a quick prerinse before they went in the Cvan park machine. (shower was a once-off lol)
I used toilet paper for the bulk of the bum wiping, and then just liners to finish up so they didn't get that much poo on them at all.
For on the road I had a Magic Washer that you turn by hand. It's kid and dingo-proof lol and worked quite well and we could use second hand water in it which is great when you're carting your own water.
As you said, find out what facilities they offer, whether you're allowed to hang outside your van (if you're vanning) or whether your only choice would be to hang inside or use communal line. Also find out what change their machines take - some cvan places have a spot you can go to get change, others don't. I was a bit scared putting my lovely nappies in the nasty-looking industrial machines lol but they went totally fine and it was only one night anyway
HTH