When BA opened their recruitment last September, it was expected to be open for 3 months. The closed it after 6 days because they had in excess of 3000 applications.
All these applications will be screened be people sat in front of computers. BA are expecting a huge amount, but if they were to receive say 30,000 in the first week, I wouldnt be surprised if they closed it after one week.
Whilst it's wise to take some time, I wouldn't leave it for a few days/weeks... You should be able to write a couple of essay questions in a day. Factor in getting someone to read over it for grammar mistakes etc and it shouldn't take longer then two days.
Just an opinion...