Not sure when this started happening. I thought perhaps it was just my own stupidity, and perhaps I had sound turned off. I tried all of the suggestions on the web. Checking sounds, banners, etc in Mail and Sound settings. Deleting my mail accounts and re-configuring. None of these solutions worked. And I don't have an Apple Watch.
What I did notice was that on a power cycle of my iPhone the first incoming email did play the sound. Then all subsequent incoming emails were silent.
The fix that worked for me:-
Settings >>> Focus >>> Share Across Devices - turn off. Now all incoming mails play my requested sound. Toggling this setting back on and the sounds continue to play on incoming mails.
I suggest a software bug in IOS
My iPhone is a 13 Pro Max model running IOS 15.7. I have seen people report the problem still occurring after installing IOS 16.
Within minutes of submitting this fix/workaround I found incoming emails no longer made a sound again. I fixed by the following:-
Settings >>> Focus >>> Share Across Devices - turn off. And leave it turned off!!! My incoming mails now play my requested sound