Short answer: Potentially February or March of 2020.

While it's not clear when recently released films such as The Lion King (2019) will be coming to Disney+, we know for a fact that company is at least open to adding their latest films to that service 200 to 249 days after its theatrical release, as seen in the chart below:

U.S. Theatrical
Release Date
Number of Days Until it Was
Released on Disney+
Captain Marvel March 8, 2019 249 (8.3 months)
Dumbo March 29, 2019 228 (7.6 months)
Avengers: Endgame April 26, 2016 200 (6.6 months)

Since The Lion King (2019) was released in theaters on July 19, 2019, it could potentially be coming to Disney+ sometime between February 19, 2020 to March 24, 2020.

Updated answer (January 28, 2020): Disney+ has added The Lion King (2019) to their streaming service on January 28, 2020.