How can I stop the timer from interrupting music on my iPhone and Apple Watch?

well presumably if you set a timer, you want to be notified somehow when it expires right?? so how do you want it to work?

I just did several tests and it all worked out in a way that made good sense to me.

Set timer on iPhone, while playing music on iPhone, using iPhone when timer expires

-music pauses and timer tone plays on iPhone with a banner notification (nothing on Watch), stop the timer & music continues

Set timer on Watch, while playing music on iPhone, using iPhone or iPhone sleeping when timer expires

-music continues playing on iPhone, Watch vibrates for timer done --- ⬅️ that would work for you right?

Set timer on iPhone, playing music on HomePod via Airplay, using iPhone when timer expires

-music continues playing on HomePods, iPhone plays timer tune (not Watch)

Set timer on iPhone, playing music on HomePod via Airplay, iPhone sleeping when timer expires

-music continues playing on HomePods, iPhone plays timer tune, Watch vibrates

If you're playing music on your iPhone when the timer expires, I don't see any way to stop it from pausing the music. You can, of course, leave feedback for Apple.. suggest when using iPhone at timer expiration it just produces a banner notification.. or the list of tones includes a vibration only option or something??

Feedback - iPhone - Apple

Or what I would suggest since you also have an Apple Watch.. set your timer using the timer app on your watch.

Another suggestion would be to find a 3rd party timer app. I just happen to have Timer+ by Minima Software.. it does exactly what I mentioned above. Since it's a 3rd party app, you get the typical Notification controls where you can turn off sounds etc. But the timer itself can be set up with no sound.. so I tested that while playing music and using the iPhone, it simply does a quick vibration and does not interrupt the music in any way.