I think the main question is how much time you are planning for this trip, and what do you intend to do with the bike afterwards (sell it? keep it for your next trip?).
Having family in the UK obviously makes buying in the UK a lot easier, but consider that 1) UK insurance seems to be ridiculously expensive in general, and 2) you would be riding a non-EU bike into an EU country. I suppose if you're an Australian citizen anyway and bound by the 90-day Schengen limit, it doesn't make a huge difference, but it may result in extra faff if you want to store your bike within the EU between trips (in the UK, I guess you would have to SORN it?).
There are places in the EU where buying a bike in your own name as a non-resident is fairly straightforward - Estonia is one of them, there are others.
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