yes the usual process will happen as follows
OP flights on their flight
2-3 days later the TPs will post. The app may indicate 'XX tier achieved")
2-3 days after that status will be updated on the app / website - may need to refresh the app or even log off and log on
No need to call. Initially if they don't post within a week then use the claim form on the website. Keep your BPs until TPs have posted in case you need to scan and send them with the claim
Although the TPs will physically post in the new year they will count for the old year. They won't be double counted because the system is based on flight date not posting date.