Mail on Apple Watch
If my iPhone is turned off and my watch is connected to cellular I can’t send or receive email.
is something wrong or does my phone need to be on even if it’s not with me for mail to work on the watch?
Apple Watch Series 7
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I'm not completely sure, but email is NOT listed in the 'If your watch is connected to Wi-Fi or cellular' section of:
Use your Apple Watch without your iPhone nearby - Apple Support
These articles also imply that you need an iPhone:
Manage mail on Apple Watch - Apple Support
Read mail on Apple Watch - Apple Support
Write and reply to mail on Apple Watch - Apple Support