The process may vary state by state but the general procedure is as follows:
When the final account has been approved, the court will order the fiduciary to distribute the assets of the estate. The court order may state the date by which the distribution should occur. If not, you may be able to request the court modify the order to include a date certain. The fiduciary then distributes the property to the beneficiaries according to the approved distribution. There should be no delay in distributing the assets of the estate after the court approves the final account.
After the fiduciary distributes the assets of the estate, the fiduciary's final act as fiduciary is the filing of an affidavit of closing with the court. If a probate bond has been required, the court will send the surety company a certificate stating that the fiduciary has complied with all orders of the court relating to the settlement of the estate and terminating the probate bond.
Disclaimer: This answer does not establish an attorney-client relationship and does not constitute legal advice. It is for general information purposes only.
"Final court date" can be a deceptive term. If it is a hearing to approve proposed distribution, and all assets have been liquidated or are othewise ready to distribute, then it is a matter of sending the assets out to whoever they belong to, which can happen in a matter of days or weeks. Each heir should sign a receipt acknowleging what they got.
If property still needs to be liquidated, then money doesn't get handed out until the liquidation is complete.