I'm not encountering issues with my Blackberry KEYone (Android).
Just did a test.... had my KEYone email the Pixel and then emailed back. Email took about 10 to 15 seconds to show up on the other phone.
Does it make a difference if you are using wifi vs cellular? Also who are you using for email? Is it gmail?
It's possible to have issues with software on any phone, even iPhones. Some of the issues could really depend on which app you are using for those services and how it's configured. Since we're all human, there is also the possibility of error as well.
Did you update the Pixel 2 XL to Oreo? Other users (worldwide) have been having issues with their phones since the update, including with bluetooth. Make sure your phone is up to date just in case.
Try using something better than myMail. I would never touch the myMail app. Too sketchy. MyMail also notes on their Play store page they use Facebook ads / user tracking in the myMail app.
Do you have antivirus on your phone? Probably wouldn't hurt to make sure you don't have some malware in there causing issues.
The hospital won't be causing any issues. When you're at work, it very well could be the building you work in that is interfering with the cellular signal reception. Concrete and steel are two types of material that will block signals.