The question is the topic title, but yeah after you've planted a bush and it's fully grown, is it possible to relocate it afterwards? or are they like trees and stuck whereever you planted them?
You can't relocate them once planted. You can dig up the bush w/ a shovel and buy another sapling to plant in the new spot you want. Same thing with trees, you may remove them with an ax/shovel, but you won't get the sapling/tree/bush thing back in your pocket if that makes sense.
Sadly, they aren't like flowers where you can pick them up, put them in your pocket, and replant them elsewhere without losing the progress. They're more like trees where you have to shovel them up from the ground and then plant a whole new sapling where you want them.