Chrome Service Worker iOS Support
Chrome for iOS do not support service workers currently(as of Apr-2020) and also the "Add to home screen" option is not available, if you are interested in that. Any offline page loading that you are experiencing should be from regular browser cache.
Here is an official documentation from Apple's web-kit page.
Update: A previous version of this post stated the Service Worker API is available in all applications using WKWebView. At this time it is only available in Safari, applications that use SFSafariViewController, and web applications saved to your home screen.
Notably, Chrome uses WKWebView as its browser engine in iOS, which don't have service worker support yet as per the above statement(which you can find in the link provided above). So the only way you can use Service worker in iOS is to use Safari browser or use one of the above supported web-views inside a native app.
Here is an article on what all PWA features iOS is still missing. Remember, this is a fast growing area. Always check if what is applicable now, especially if the documentation/article is getting older.
Here is an adapter to help debug in Chrome dev tools for pages loaded iOS - webkit/safari. Hope it becomes as easy as Android Chrome debugging one day!