Kubernetes have auto-healing

You can let the POD crash and Kubernetes will auto re-start them soon as get the sufficient memory or resource requirement

Or else if you want to put the wait somehow so that deployment wait and gradually start one by one

you can use the sidecar and use the POD lifecycle hooks to start the main container, this process is not best but can resolve your issue.

Basic example :

apiVersion: v1
kind: Pod
metadata:
  name: sidecar-starts-first
spec:
  containers:
  - name: sidecar
    image: my-sidecar
    lifecycle:
      postStart:
        exec:
          command:
          - /bin/wait-until-ready.sh
  - name: application
    image: my-application

OR

You can also use the Init container to check the other container's health and start the main container POD once one POD is of another service.

Init container

i would also recommend to check the Priority class : https://kubernetes.io/docs/concepts/scheduling-eviction/pod-priority-preemption/