Normally, joining dates are indicated in advance to the candidate and later formalized in letters of appointment. When you do not accept the offer received by you in full, it becomes a conditional acceptance, and therefore it means you have not accepted as such. Your conditional acceptance is not binding on the firm. If the firm accepts your condition, you are lucky. Otherwise, if you peruse the letter, there will be a clause that (a) if the offer made by the firm is not accepted by the candidate within a defined date mentioned in the letter of offer and/or (2) in the event of the failure of the candidate to join on or before the specified date, the offer made to the candidate will be deemed to have been automatically withdrawn or canceled.
We have to be very careful in handling the letters of appointment received by us. Every clause has to be read carefully, understood, and acted upon without fail.
I do not know why you could not have talked to HR over the phone before you sent an email requesting a change in the date of joining. Even now, you should make an attempt to talk to HR over the phone and elicit their response urgently.
Regards,
R. Sundararajan