Ideally, the relieving letter should be issued to the resigned associate immediately upon their exit. The ideal practice is to complete the Full and Final (F&F) settlement on the last day of the associate's employment with the establishment and relieve them with an F&F settlement copy, relieving letter, and/or experience letter (or a combination of both). Since the F&F settlement copy would have the associate's signature (as an acknowledgment), the establishment can pay out the dues to the associate within a couple of days (unless it’s a termination case wherein you need to settle their dues immediately). In case the associate has to pay back to the establishment, they could be asked to submit a cheque on the last day of their employment, subsequently followed by the issuance of the required relieving and/or experience letter. This will enable them to submit the same to their new employer.
But you need to have your HR process in place to get this process completed on time. Moreover, the "Mr/Ms Responsible" (for this activity) would have to be quite proactive and assertive to get the deliverables in time, since it would require a lot of coordination with different departments/Finance-Accounts/etc. It's an excellent HR practice (since it would speak very highly about the organization's excellence and professionalism) but it has to be driven from the top and that's where the role of the HR Head comes into play.
I know of a few MNCs that religiously follow this standard process, but these are companies with excellent HR policies and practices. However, I have seen many other organizations that either settle your account after a few days or even later to the associate's exit. In most of the Government (both state Govt and central Govt.) organizations, you would find the relieving/settlement process getting done days/weeks later.
Best Regards,
A.B.