I had to do some manual updates to get the JL ticket booked via CX to show up on the JAL app.
The problem with me was the CX/Asia Mile membership number was transferred directly to the membership number field on the JL side where it supposes to have a JAL Mileage Bank number. This was causing the booking not able to retrieve on the JAL app.

Steps I took to fix this.

1. Go to jal.co.jp website. Change language if necessary.
2. Scroll down to Manage Booking
3. Select International Flights
4. Make sure you are not logged into JMB or click on Continue as JMB member. Under the Non JMB Member, search for your flight, use your Booking Reference Number from CX in the Reservation number field.
5. Once the booking is found, scroll down to the Passenger & Contact information section and click on Related Services
6. In the Passenger Information section, the CX membership number is automatically filled in for the membership number under my name. Update the Membership number to your JMB number, and select Japan Airlines as the membership program. Scroll down and save the change.
7. After 15 minutes or so, the booking showed up on my JAL app.