The processing for JWST's alignment is done on Earth. How long does it take for a signal from Earth to reach the JWST?
Almost 5.1 seconds, plus or minus around 1 second.
Here's a daily plot (at midnight) of the light travel time from the JWST to the centre of the Earth, courtesy of JPL Horizons, using a script derived from the one in this answer. There are more graphics & scripts related to the JWST here. Times are in TDB.
The light travel time to a location on the Earth's surface has a small extra variation due to the Earth's rotation. And of course, you can't get a direct line of sight signal from the JWST when you're on the wrong side of the Earth. ;)
Here's an hourly plot for today of the distance from the JWST to its control centre, the Space Telescope Science Institute, which is located on the Johns Hopkins University campus. I used longitude -76.622987, latitude 39.332887, altitude 0.073 km as the coordinates.
Here's my plotting script, running on the SageMathCell server.
According to NASA, the JWST is 1.5 million kilometers away, which is about 5 light seconds.
For comparison, the distance to the Moon is about 1.3 light seconds, Mars ranges from about 3 to 22 minutes, and to the Sun is 8 minutes.