That's weird. Don't they keep records of their past students?
Anyway, as far as missing the deadlines is concerned, for most of the schools, you only need to "self-report" the transcripts. So, the delay in getting the official transcripts shouldn't harm your applications. Just make sure that the official transcripts reach the schools after you apply and get selected, or get called for the interview .. depending on the target school's policies.
Neo, that is the process followed usually in India.
In order to get my transcripts, I had to take photocopies of my originals, and go to my university examination branch and submit them there.
Going in person is compulsory. The person can be a relative or friend also. Submitting photocopies is compulsory. The examination branches are too disorganized to take photocopies and take signatures and send as transcripts.
I did n't even rely on them to send the transcripts through courier. I requested my sister to go and collect transcripts. We just can't imagine how crappy things can be here in public universities.