Run the Laravel queue worker as a process using supervisor. The included configuration file has been minimally changed from the example in the Laravel docs.
Copy Dockerfile and laravel-worker.conf to .ddev/web-build and restart using ddev restart.
Keep in mind that using this approach will modify your web image, so read the caveats in the DDEV docs under "Customizing Docker Images" > "Adding extra Dockerfiles for webimage and dbimage."
Copy config.laravel-worker.conf to .ddev. When the container is restarted, the queue worker will run in the background with no change to the Docker image.
Contributed by @karlshea