You could simply build your own shortcut on the iPhone, with “start Timer” (or something like that, I only have the german UI) and maybe a tone, that is played between each timer.
You may also consider some break time in between…
I recommend checking out MultiTimer. I’ve been using it for years when practicing and teaching yoga and it continues to serve me well. Multitimer includes a wide variety of timers, including the “interval” timer, which should work well for the use case that you mentioned.
I’m the developer of this app, the android version is the top Interval Timer App on Google Play and I added an iOS version around a year ago.
It is ad-supported, but has few ads and none of them during the workout. There is also a one time purchase to remove the ads for ~2.99USD
here is the App Store link: Interval Timer Plus
I started this app out of personal interest for workouts.
It has two modes: simple and advanced.
Simple mode works basically like the hardware round timers you are used to from boxing: number of rounds, work, rest
Advanced mode allows a lot of settings, but requires more time to set up.
One of these options is having multiple intervals of different length in a sequence, like the one you are looking for.
I’ve setup an advanced preset for you, I added 8s before “heavy bag” for putting gloves on and 5s after to take them off and move to the battle ropes. This repeats 3 times with 10s rest in between.
Clicking the link below should take you to the AppStore and after installation open the preset for you to save to your device after that you can edit it to adjust it to your needs.