JediDude I was in this situation a few years ago while I did my training.
Strictly speaking you will not be able to claim JSA(jobseekers allowance) as you are not available to work whilst training. As for council tax benefit, as has been posted here, why not give your local job centre a call they will advise.
Your wife may possibly be able to claim JSA but she then must be available to work and there are childcare allowances available. She can also work part time and still claim JSA, up to either 16 or 24 hours a week I believe. The benefit paid reduces the more part time work she does.
Stay at home mums cannot claim because they choose to stay at home, it is of little consequence that you have paid vast sums into the system. The system is set up to encourage people back in to work, if your wife finds part time employment so does someone else ie: the person who takes on your childcare.
The day you finish your training and are finally unemployed JSA is available, believe me the jobcenter will not be able to help you with any aviation related work
