In Australian English, which has a slang meaning of "root" which is best avoided in a professional setting,
This is not restricted to Australian English. Root can mean "penis" Irish English, is attested in British English since the 1840s (and as late as the 21st century) and appears in a Canadian play of the 1970s. It can also mean "copulate with" in Ireland, and I'm sure that applies elsewhere too.
is "root access" acceptable in a professional setting?
Of course it is. As would any other reference to the concept of root in computer science and IT (and there are several such concepts), agriculture, horticulture, along with ways it might be used in every field.
There isn't even the shade of an issue here. If someone sniggers every time they hear the word "root" or can't resist double entendres like "I'd love to give them my root access" in situations where it would not be appropriate, the problem is not with the technical jargon.