Will my childcare claim be processed if I have put the last few weeks childcare of one academic year onto the childcare application form for the next academic year?

Yes, these will be processed and added onto the payment schedule for the correct enrolment period. This will only be relevant where the new academic year starts midway through a month.
 
Example
 
Your new academic year begins on 22 September 2014. You have entered childcare costs from 1 September on your 2014 / 15 application.
 
These weeks childcare will be added to your payment schedule for the 2014 / 15 enrolment period.