Is there an app / website thats a HIT timer but also chooses the exercise for you?
Ive Googled this and look on app stores but I can only find timers only. With some you can set the exercises you want to do but non pick them for you.
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The YAYOG app lets you define your custom workouts. They have multiple exercise types, including Tabatas, which are HIIT intervals with 20 s load and 10 s rest. You could - for example - create a Tabata workout with 10 rounds (3 min) of push ups and 10 rounds of squats.
You could have a look at Freeletics. There are many workouts you can choose from, most of them being HIT workouts based on bodyweight exercises. I don't know, if there actually are HIIT workouts, but it's definitely conceivable. There are some free workouts, but for most of them you'll need to have a subscription.
You can start the workouts both from the website and from the app. Both will keep track of your times and guide you through the workouts.