My understanding is that, usually, the target company will gain...

My understanding is that, usually, the target company will gain ATO approval for rollover tax relief which should mean that there will be no capital gains tax event triggered through the merger and so normal tax provisions will apply: tax will be payable on the profit generated when the Energy Fuel Shares holding is subsequently sold. This would apply if you trade the sale and proceeds through the international trading facilities of an Australian broker account. If you choose to leave any sales proceeds in US and trade there then I'd expect US Tax will also get involved.
I expect BSE management are investigating options here and hopefully will provide detail in the forthcoming scheme of arrangement document