Selection Process and Candidate Management
I think you are not involved in the selection of candidates in the many rounds of interviews. It's one of the primary duties in the recruitment process that the selection committee shall submit to HR a panel of selected candidates in order of merit. The top-ranked one is considered the successful candidate, and you send your 'Offer' letter incorporating, among other conditions, the i) last date to submit/return 'acceptance of your offer' and ii) last date for reporting following the receipt of 'acceptance of offer'.
If (i) acceptance of the offer is received and the offered candidate is willing to join, you have to wait until the 'last date for reporting'. If he/she doesn't report by this date or no request for an extension of 'joining time' is made, you should assume that this candidate is not going to join. Following this, you should initiate action to obtain approval from the competent authority to issue an alternate offer to the next, the 2nd ranked candidate. Once this is done, there is no waiting for the 1st ranked one who is declared as 'no show', and he/she should be blacklisted from applying to any of your vacancies.
In the process, if the 1st ranked candidate applies for an extension of joining time, you may consider the request on merit or decline the extension sought, depending on a decision by the competent officer(s). This should solve your problem. Therefore, it's a step-by-step process. So ensure the interview committee submits its 'panel of the list of candidates' interviewed, which should help you to take appropriate action as I described. I don't feel you should remind or engage in any conversation with a 'candidate not interested' to join. But sometimes the timeline may not be adhered to if the process of BGV and verification of credentials, relieving letter submission in respect of selected candidates gets delayed for some reasons.
There is literally nothing, no device to read one's mind. Sometimes the 'interested candidate', really, need not wait until the elapse of the 'last date to report' when he/she is free to join with all formalities accomplished. It is not advisable to call everybody when an 'offered candidate' fails but should abide by the list of the panel in that order. When such things happen, there is nothing to worry about as I hope there are plenty of talents still left in India.
One suggestion I could make is to try 'freshers' who are found to be brilliant, promising, and employable. Also, try 'campus interviews' for your requirement.