You can use a DONE4 ticket. That's a Oneworld round the world ticket. It will cost you about GBP7k.It has no ticketing deadling and great flexibility; the only restriction is that you will need D class availability which may be a bit tight close in.
Eg, route
LHR-JFK-LAX-SYD-SIN-LHR
Of course, many would wince at so few flights in such a lovely flexible ticket but you want fast, not lots of stops or tier points.
Otherwise you're going to find it expensive to buy one way tickets to do this especially at short notice, but a travel agent might be able to do something. If you can use the DONEx ticket, it's simpler.