Hi Ben,
Heres a couple of firms you can trade the shares with 24 hours with but be wary firstly of the margins. As you will be trading out of hours they may unexpectedly raise the margins quite high and find a reason to stop you out prematurely which could cost you.
City Index UK - Spread betting and CFDs on financial markets
Cantor Index - 24 Hour Betting
Spread betting with CMC Markets 24 Hours a day
Ideally i would if i were you look for a share that is multi listed if you can on all the major exchanges like Asian, UK and American stock markets then that way the SB Company will be able to take the proper market traded price off their Bloomberg and you are not getting an indicative price the SB company makes up. Im not sure if you can find a mining share on all exchanges though. A call to the SB Co's above should give you the info you need.
The other option which im guessing at is you could open an account with Interactive Brokers who you can trade on all the major exchanges with. You may be liable to Pattern Day Trader rules with the US. Im not sure about Asian markets but im sure someone will better knowledge of these areas will add some more shortly.
Ged