Stake callings are entered in stake MLS, and will then be transmitted to CDOL within a few minutes from the time they are transmitted from stake MLS. Once they are in CDOL, they are immediately available in the Stake Leaders and Organizations section of the Directory under "Stake Information".
The wards are not involved in this process (and should not be entering stake callings -- that just leads to a lot of duplication and maintenance problems).
Note however that for this weekend, there is a major CDOL upgrade underway, so there will be a delay before callings entered in stake MLS are visible in either CDOL or the Directory. They won't appear until sometime on Monday; see
CDOL 3.0 Upgrade this weekend; no changes will be made to CDOL until Monday morning.
Questions that can benefit the larger community should be asked in a public forum, not a private message.