Updates to a new calling take but a few seconds, maybe fifteen minutes at the most. You can see this as a general member in the directory generally within fifteen minutes of an MLS update and send/receive. Permissions may take a day, or maybe two, after the update to CDOL as russellhltn stated.
If the calling is an assistant ward clerk then additional tools will be available on the LDS.org tools menu. The four most obvious new tools for an assistant ward clerk are Clerk Resources, Leader Resources, Missionary Recommendation, Church Directory of Organizations and Leaders, and Facility Issue and Reporting. An assistant ward clerk - membership will see the same things as an assistant ward clerk. An assistant ward clerk - finance will see no changes at all because that position has not been granted any rights beyond general membership rights on LDS.org yet.
So if you see the additional tools then the calling has been processed. If there are no additional tools then the calling has not been processed.
In LDS Tools you must do a sync to get an update to the database. If you have not been prompted to enter a pin after a sync then the position has not been updated in the database for LDS Tools. I would make sure the calling position is reflected on LDS.org, the tools, and in the Directory, before concerning yourself about LDS Tools issues.
The MRN is visible in the Directory, in any display of membership data in Leader and Clerk Resources, and in membership data displays in LDS Tools for the assistant ward clerk and the assistant ward clerk - membership.
JD Lessley
Have you tried finding your answer on the ChurchofJesusChrist.org Help Center or Tech Wiki?