Apple Watch with LTE connectivity can do a lot of things without connecting to iPhone, but there are a few nuances in their work with calls and SMS / MMS.
SMS / MMS messages
According to the doc Apple about using Apple Watch without iPhone:
To receive SMS / MMS messages to your Apple Watch Series 3 (GPS + Cellular), your iPhone must be turned on and connected to a Wi – Fi network or mobile Internet, but it is not should be near you.
In other words, if your iPhone is off, in flight mode, or not connected to the Internet, SMS / MMS messages will not appear on your watch, even if it is connected to the LTE network.
The reason is that SMS and MMS work differently from phone calls and iMessages.
CNET editor Scott Stein made it clear that your iPhone could be on your bedside table at home 50 miles away, and you can still text SMS / MMS messages on your watch if your phone is connected to the Internet.
IMessages on your watch will always work.
Phone calls on the watch can always be received when connected to LTE.
According to Apple:
If your iPhone and Apple Watch Series 3 (GPS + Cellular) are both connected to a cellular connection, then the watch can perform all its functions, even if iPhone is not with you.
Tip: Turn on Wi-Fi calling to receive and make phone calls on your watch even when it disconnects from LTE and connects to Wi-Fi.
If you use the watch without iPhone, then notifications from third-party applications like Instagram, Twitter, Facebook, etc. will not come to them .
'Apple did not say anything about this change in the future,' writes CNET. To receive notifications from third-party apps, your phone must be nearby as they arrive via iCloud.
In addition, apps must support the watchOS 4 API to work remotely from your phone.
For example, the Slack app will not work on the new watch because it uses older APIs. Native apps Apple like Maps, Mail, Messages, Weather, etc. will always work.