It depends. There is no set period for which an employer cannot recruit after redundancies have been made. However, depending on the facts, recruiting for the same role as that of the ex-employee could compromise the fairness of the dismissal.
Genuine redundancy
The dismissal could be unfair if the role is the same as that of your former employee because that would suggest that the redundancy was not genuine and that you have the same requirement for employees to do work of a particular kind.
Recruitment freeze as an alternative to redundancy
Even if the role is different, your former employee would question why you didn't freeze recruitment as an alternative to redundancy. A dismissal is more likely to be considered fair if the employer has considered whether it is possible to avoid or reduce the need for dismissals. Freezing recruitment is one option alongside:
withdrawing new job offers
deferring new joiners
reducing agency workers
redeploying or seconding staff members.
If challenged, you should show that these steps would not have saved enough money quickly enough, so you considered and rejected them.
Suitable alternative employment
The dismissal could also be unfair if you should have offered the new role to your former employee as alternative employment. That would be subject to how the new role compares to the redundant role, whether it could be deemed to be the same or similar or that there is a crossover of skills that you should have considered.
Changed circumstances
Of course, you could have a genuine redundancy situation and a new role. You may have won new business that you were not expecting, or a colleague may have resigned.
Re-employment
What if you wish to re-employ someone after they have been made redundant?
If the redundancy was genuine then the employee's statutory redundancy payment at the time of the original dismissal should not be affected by a later re-engagement, unless the offer of the new employment was made before the original employment had ended.
However, if the duties are essentially the same and there is no apparent commercial justification, HMRC may argue that this was not a genuine redundancy and re-engagement. Suspicions are likely to be aroused in a case where there is a very short gap between the termination and the re-engagement.
If you offer new employment before the original contract has ended, special rules apply and continuity of employment will be preserved. The new position must take effect within four weeks of the termination of the old contract. In this situation there would be no entitlement to a statutory redundancy payment.