It varies. If you booked a confirmed (existing) reservation as opposed to a new reservation, it can take a while. If you booked a brand new reservation, it shouldn't take more than a day or two.
That's because confirmed reservations have other names attached and the owner must contact member services to make the change. That can take several days for member service to get around to accomplishing the change. Until then, you won't get a confirmation because you will not be able to link it to your own
MDE without the matching names.
Now, if you only put in a request and have not actually booked/paid for a reservation, that would be dependent on a variety of factors and could take weeks or even months.