I currently have access to a Whoop, Oura ring, and Apple Watch.
Given all 3 of these apps log “wakeup time”, is it possible to automatically trigger a routine via zapier/integromat/ifttt/shortcuts based on when one/any of these devices “detects” that I have woken up?
How technical are you?
You could use Whoops developer portal to trigger a webhook for live updates. Oura has a similar functionality as well.
Apple Health iOS doesn’t have a REST API so you would need to be pretty savvy with the dev toolkit or use a tool like the Auto Health Export app.
Love this question. I am not sure about the automatic routine based on when you wake up.
All I can speak for is Oura as I have been able to log the info but not use it immediatley
[How to automatically export Oura ring data to google sheets every day.]
(How to automatically export Oura ring data to google sheets every day. - YouTube)
In my experience I had to track yesterdays data because sometime the Oura Servers are delayed.
My next thought to share was you can set up a shortcut with Siri and simply say “start morning routine” but that means you actually have to say something and it doesn’t happen automatically
As for apple watch it too is not so accurate, I will get up and go to the bathroom and 15 minutes later it asks me if I am not sleeping any more.
Not sure what your intent is here so if you have more questions, let me know 
Normally you can set a shortcut to run on your phone each time a focus mode is turned on or off. However, you can’t for the sleep focus mode for some reason.
Is there a particular reason you want to trigger the routine automatically instead initiating it yourself? Do you wake up on an alarm or without an alarm?