The death of an employee is never an easy thing to deal with, on either an emotional or business level. The death may have come as a shock to you and to the rest of the team. It's a juggling act – on the one hand you need to give yourself and others time to come to terms with the loss, on the other, you need to keep the organisation moving.
How soon you replace an employee really comes down to time pressures. In some roles – particularly senior ones – time is rarely on your side. Some roles need to be filled quickly, but that still means recruiting with tact and giving people time to adjust. You should leave introducing a replacement as long as you can, within reason, being mindful of both the urgency of the work and employees' feelings.
At other times, it's a little less pressing. If that's the case, you should inform people that you won't be starting to look for someone for two or three weeks. Again, it's a fine balance. It's not fair to leave people shouldering extra work for too long, and the sight of an empty desk can be more upsetting than the sight of a new face taking the place of the deceased.
The best thing you can do is keep people informed, so that nothing comes as a shock to them. Suddenly finding a new replacement colleague in their midst might seriously upset some people, so communication is key – otherwise it's not fair on the new recruit or your current employees. Be both time and emotionally sensitive as you proceed.
It all boils down to time pressures
If it's urgent, start a recruitment drive in a couple of days
If it's less urgent, leave it a couple of weeks
Always keep people informed
Be sensitive to both time and emotions