You are not legally obliged to follow your grievance procedure for an individual who is no longer employed, but it is good practice to do so and may help you if the matter is taken further by the former employee.
There are wider issues in the example you have outlined above. If a member of staff submits a formal grievance, you should not terminate their contract until the matter has been fully resolved. The correct course of action would have been to extend their probationary period, such that you retain the power to terminate their contract without notice at a specific date after the end of the grievance process. Depending upon the timing of the different actions, a tribunal might take the view that your decision to terminate was wholly attributable to the submission of a grievance. That is unfair dismissal.
Check your own policies and the individual's contract of employment to ensure that all relevant actions have been taken. Did you provide appropriate feedback to the employee during their probationary period, identifying any shortcomings and offering support for improvement? Are any of the perceived shortcomings a result of a protected characteristic? If termination is your chosen option, ensure that any final settlement includes the statutory entitlement to a notice period and accrued holiday entitlement, all of which must be paid.
Looking at the wider issue, a grievance from an employee often highlights wider issues within your organisation that need to be addressed, whether in relation to the current postholder or their successor.