There are two separate issues here, I think. So you are on working holiday visa in Australia, as is your wife. You are an Estonian citizen and want to bring your non-EEA wife to UK with EEA family permit. But both your Australian visas are expiring soon, and if you apply for EEA permit now, it may not be issued until either or both visas expire.
First, EEA family permit can be applied at any British diplomatic post that issues visas. So why don't you go back to Estonia and apply there? It creates problems with Australian immigration if you overstay your visa, and you may not be let back in just to collect your EEA permit, as they may suspect you may try to work and stay on illegally. Also you are without your passport while UKBA process your application, which precludes leaving Australia.
Secondly, applying in another country while staying as a visitor is possible, but each consulate has its own procedure and processing times, which can vary hugely.You need to find out before you travel there.
These are the options as far as I can see.