docker-compose healthcheck does not work in a way it is expected for making container a run first and then container B

The two examples are based on the condition form of depends_on which is no longer supported in compose version 3. So, unless your docker-compose version is <3 the healthcheck will not help you much. The healthcheck sets the status of the container (starting, healthy or unhealthy) but docker-compose does not wait until backend container is healthy before starting the app-test. There is a detailed explanation about how depends_on works in Control startup and shutdown order in Compose

As a side note, the healthcheck in your compose file sets the status of the app-test container and not backend.

Therefore to control when the api-test can start, you have to wrap the service command of the container. For your particular case the following will do the job:

bash -c 'while [[ "$(curl --connect-timeout 2 -s -o /dev/null -w ''%{http_code}'' https://backend:3015/readiness)" != "200" ]]; do echo ..; sleep 5; done; echo backend is up; <service_command>'

It tries to connect to backend every 5 seconds (the connection timeout is 2s). When the received HTTP status code is 200 OK the loop ends and it executes the <service_command>

The relevant docker-compose part:

  api-test:
    restart: always
    command: bash -c 'while [[ "$$(curl --connect-timeout 2 -s -o /dev/null -w ''%{http_code}'' uds-mock-server:4000/readiness)" != "200" ]]; do echo ..; sleep 5; done; echo backend is up;npm start'
    depends_on:
      - backend
      ...

Hope this helps.

it's not localhost connection string.

It should be service name of backend container :

test: ["CMD", "curl", "-f", "http://backend:3015/readiness"]