Hi GroovyOne,
I didn't know we had any FIFO workers in the UK, or Europe for that matter - I'm surprised anywhere is remote enough to make it worthwhile (apart from offshore rigs of course, but we don't use the term FIFO for them although obviously that is the only way to get there!).
Just to pick up on a couple of things about the high schools, Wimbledon and Putney do indeed have a higher entry cut-off than Surbiton, but that's not really relevant - the problem that you will have with any school is that you have missed the boat for September 2014 11+ entry - there are thousands of SW london girls queueing up for places in these schools that sat exams for them at the beginning of the year who are waiting for letters awarding places or waiting list places in February/March.
It is sometimes possible to jump the queue, particularly for expats where schools are often sympathetic with the difficulties, but you would have to approach the schools directly to see if they were willing to be flexible.
Surbiton is a nice long established and straightforward school, but it is NOT part of the Girls Day School Trust unlike Wimbledon, Putney, Sutton and Guildford Highs which are all GDST schools. The academic bar is lower at Surbiton, and there are more places available for "outsiders" (all these schools are fed by linked junior schools which generally account for about half the places), partly because it is a bigger school.
Its sporting facilities and the timetabled use of them are not likely to impress someone coming from Australia (except for rowing), but the first teams are reasonably successful.
Surbiton is unusual in having and associated boys prep/pre-prep (4-11) which is relevant to you.
Surbiton is a great place to live with a fast and frequent train service: it can be quicker to reach parts of London from there than from Fulham: Surbiton to Green Park (station to station) is about half an hour. However this means that it is very popular with limited rental property available. As you go further out along the same train lines (Esher, Chobham, Weybridge) house prices/rents stay similar but with better availablity (currently) and you get more space for your money.
Back to schools - there are independents in SW London that are more likely to have place for your daughter in September but you are unlikely to find them by starting at the "aspirational" schools and working down - this is where Gabbitas are invaluable. The Hall is one of them, Marymount is popular with expats so often has spaces but probably not at 11+, Canbury and Hampton Court House are quirky schools often used for a couple of years by expats...