You could fly Singapore one-way and wait for the availability on the other leg to open up and purchase that as a separate one-way ticket - I don't see a reason to necessarily book it as a round-trip. Singapore also has very modest change fees, so you may want to book that as your primary option and cancel if necessary.

You should be able to find some United availability to Australia/NZ in business, if that is acceptable. I wouldn't go so far as to say AA and Qantas availability from NYC-Australia is nil, but it's awful close. I've found and flown Qantas LHR-MEL in the past, which might be another option, and then there should be plenty of NYC-LHR awards as a separate ticket.