How to assign a dynamically updating date to a user based on a related field on their profile
You should be able to do those things without coding.

You can use the entity reference module to add a field to the user's profile that references the relevant delivery route node.

Then you can use the rules module, along with the rules conditional module to do the rest.

Have a rule that on user registration, uses the "Fetch entity by property" action to load the node that has the matching post code.

Then use the "Set a data value" action to set the value of the entity reference field for that user, based on the fetched entity.

Have another rule for when a user profile is viewed that uses the "Fetch entity by ID" action to load the delivery route node for that user.

Then add an if conditional that checks the entity type and content type, or use "Entity has field" and check that the date field is present. This step is required for you to be able to use all the fields on the fetched node in later actions.

Then you can use the "Show a message on the site" action to show a message to the user, in which you can use tokens to show the value of the date field attached to that delivery route node.