Power washing a bicycle I'm moving from the UK to Australia later this year, and shipping most of my stuff, including two bicycles (simply hybrid bikes, nothing fancy).

Australia has strict biosecurity rules about what you can bring into the country, including cleaning requirements for things like bikes. They want to avoid you bringing in any foreign seeds or bugs that could cause problems.

I have a cheap power washer that would certainly do a good job of cleaning the bikes, but I'm concerned that it could blast water into bearings and whatnot, potentially causing issues.

What do you recommend?
Power washing will clean your bike, followed by closely after forcing you to replace all your bearing compartments.  This means your headset, bottom bracket, pedals, and wheels (hubs, freewheel or freehub).  

Do not powerwash your bike.  At all.  With all the effort it would take to properly "cover" all the bearing compartments, then had washing all those areas after, you could have just cleaned the entire bike by hand properly.