Request for Technical Guidance on Real-time Delivery Date Updates

I am writing to seek your expertise regarding the implementation of real-time delivery date updates for products in my online store. Below are the key details of my request:
My store contains multiple products, ech one need to update delivery date.
I need to : updated delivery dates in real-time (instantaneously) when:
A user opens a product page. A user opens cart and views the cart page, The delivery dates are pulled from an external desktop application that is successfully connected to the store via API (I already have a valid token and proper API connection established).

Desired Functionality:
Product Page Load Event: Update and display the product’s delivery date when the page loads.
Cart Page Load Event: Update and display the delivery date for each product in the cart.
Show the latest delivery date among all products (for order fulfillment).

Technical Inquiry:
Could you please advise on the best approach to:
Trigger delivery date updates during these two events (product page load and cart page load)?
I would appreciate code examples or recommended endpoints if available.
Thank you for your support. I look forward to your guidance on implementing this functionality seamlessly.