They are synonyms (as RAE says in the definition you give) so in my opinion they are interchangeable. They are all adverbs of manner and they all mean that there is only one way of doing it, it's just in one thing or that you want it without anything else. So these sentences would mean exactly the same to me.
Sólo/ Solo hay una forma de hacerlo.
Solamente hay una forma de hacerlo.
Únicamente hay una forma de hacerlo.
El canguro vive sólo/ solo en Australia.
El canguro vive solamente en Australia.
El canguro vive únicamente en Australia.
Sólo/ Solo quiero un café.
Solamente quiero un café.
Únicamente quiero un café.
As solo is shorter it will be probably more used than the others in oral speech but any option is ok.
Just take care that solo (when it doesn't have the accent mark) can also be other things apart from an adverb of manner, so it wouldn't be interchangeable in the other cases. For example it wouldn't be possible in these sentences:
Está solo en casa.
Quiero un café solo (sin leche).