When will the Blood Oxygen app be available on Apple Watch Series 10?

Note also that none of the official specifications or feature lists for US-sold versions of the Apple Watch Series 10 make any mention of blood-oxygen testing. In fact Apple specifically lists the countries where that feature is available and it is shown as not available in the US:

watchOS - Feature Availability - Apple

So you have not paid for something that was promised that you haven't received. When or indeed if that feature will ever be available in US-version Apple Watches, none of us here can say.

Some more facts:

• As the sensor, which is the subject of the alleged patent infringement, is disabled by OS on models ending in LW/A, no other app will work.
• The sensor is physically there and could be reactivated, either after the case is settled or after 2028 when the patent lapses.
• Apple Watch is not a medical device and people with real pulminory disease such as COPD or asthma are urged to use a finger oxymeter using "through" LED light, and not reflected LED light which is not accurate enough for you.

when the court case is settled, who knows when.

Feel free to send your feedback directly to Apple Feedback - Watch - Apple

Excellent details. I have the US watch series 9, with the Blood Oxygen app before it was "banned". I'll have to wait till 2028 to get a new watch.

With 20% - 80% charging, my total capacity is still at 97% after 1.5 years. I'll just keep this 9 until Series 15.

The watchOS software can not be purchased by the user, and Apple can change the availability of features if needed.

Have a look at the Software Licence Agreement for watchOS