There's nowhere near enough salt in washing up liquid to cause issues when used on anything. It's an old wives tale.
It's like saying you can use grease to wash the bike because there's soap in it.
There's also virtually no danger of using a hose pipe on a bike. It might have been an issue decade's ago, but modern seals on bike hubs etc can easily withstand that sort of water pressure. Even a pressure washer is little danger to a bike unless used by an idiot.
My own regime is to wash the entire bike with lots of washing up liquid to make it more soapy and warm water. Cold water works too but takes longer to loosen up the dirt apparently (I've never really timed it) and freezes your hands in the winter.
For the chain, chainset, cassette and dérailleurs, washing up liquid on it own makes a good enough degreaser.
A common household scrubbing brush used to agitate the oil.
Do the drive train before everything else or you'll find dirty oil spots splattered everywhere.
Rinse the bike with a hose and dry either with a towel/tshirt/frilly French knickers etc or let it drip dry. I like to bounce mine on its wheels a couple times to shake most of the water off.
Along with the chain its helpful but not entirely necessary to drop a drop of oil on the derailleur pivots points, leave a while then wipe the excess oil off. Same with the chain. Don't soak it in oil but put it on sparingly and wipe any excess off. Oil attracts dirt.
A bit of polish makes it all nice and shiny but does little else. It's worth it imo.
