If you use airmiles it will costs you significantly less than cash to get into the business cabin and since the introduction of Avios Booster you can accumulate avois very quickly. Off peak to Sydney will cost you 290,000 avios plus £1,500 in business on BA for 2 tickets with a 241 voucher.
Unfortunately there's no great sign up bonuses on credit cards at the moment, but here's an idea of how you could get there:
1) Sign up to Amex Platinum card and spend £6k in 3 months to earn 6,000 avios and 40,000 bonus avios
2) Use Avios booster to add another 138,000 avois
3) Refer wife to BA Amex Premium card (around 10,000 referral bonus avios from memory). Cancel platinum card to get pro rata refund
4) Spend £10k in a year (including £3k in 3 months) to earn 15,000 avios and 25,000 bonus avios, and a 241 voucher.
5) Use avios booster to add another 56,000 avios
Total cash outlay
Amex platinum membership fee £180
Amex BA fee £250
Avios boosting at 0.92p £1,785
Flight fees and taxes £1,500
Total £3,715 or £1,856 per flight
Your challenge may be spending quickly enough if you want to go this year. Also if you plan to depart from Heathrow, getting reward flight availability this year probably isn't realistic. There is more availability from regional airports or you could defer the trip until next year and book the seats 1 year in advance when they become available.
You would also need to consider eligibility for Amex aldign up bonuses if either of you have had an Amex in the last couple of years.
