This app was mentioned in 8 comments, with an average of 1.75 upvotes
You want to look for an interval timer app. They're meant for HIIT training programmes but should work for your requirements as well.
I've been using this one for a few years now...
https://play.google.com/store/apps/details?id=com.ihunda.android.hiit
This one will do what you ask, but I think it only allows for one set interval... Such as you can do 1 minute work, 1 rest, but that's it for the workout (up to as many reps as you want though) You can't do 1 minute work, 1 minute rest, 2 minutes work, 2 minutes rest.. etc...
Unless I'm just dumb and can't figure it out.
Has no problem with my music in the background, has a 3 second chime countdown before an interval ends.
Interesting point. That makes sense. Any idea how long it would take to do a reskin of a relatively simple open source app, assuming the developer does the design from scratch? This is the app I'm talking about specifically: https://play.google.com/store/apps/details?id=com.ihunda.android.hiit&hl=en
Do we have the same apps? I did not pay for this. It's on Android. It is this one: https://play.google.com/store/apps/details?id=com.ihunda.android.hiit&hl=en
Either way, there are multiple HIIT apps on both Apple and Android markets, if there are no presets, just set the timer to intervals you are comfortable with. 30 seconds on, 10 seconds off. 60 seconds on, 30 seconds off. Whatever works for you, you can configure them anyway you like.
What you want is a HIIT timer.
Here's one - https://play.google.com/store/apps/details?id=com.ihunda.android.hiit
I use an app on Android.
If you're on Android, I use https://play.google.com/store/apps/details?id=com.ihunda.android.hiit . I don't know if it's the best, but I've been using it for years now and it meets all my needs