What is the status of a non-cloth covering, such as putting your head on your watch?
All covering which is not part of the body is good for covering, even non cloth see Magen Avraham OC 131, 1 sk 2, because there are good chatsitsa, (as not part of the body) :
(ב) להטות: טעם ההטיה כמ"ש ס"ח ובהגה"ה ומ"מ נוהגין להפסיק בבגד דיד לא חשיב כיסוי דמין במינו אין חוצץ כמ"ש סימן צ"ח:
The hand is not regarded as covering because he is from the "self species" as the head.
A watch is "from an other species", so it can cover the head.
What is the size of the watch? Seems perhaps too small. .
See Gemara Berachot 24b at the end of the Amud and see Bet Yosef OC 74 below:
וכתב ה"ר יונה בשם רבני צרפת שאם האדם מחבק גופו בזרועותיו דיינינן ליה כהפסקה מבגד ומותר ואפילו לפי מה שכתבתי בסמוך בשם אור זרוע ותרומת הדשן שאם מכסה לבו בידיו כדי שלא יהא רואה את הערוה לא חשיב כיסוי אפשר דמודו בהא דשאני כיסוי ביד דכיון דלב ויד חד גופא אינון אין הגוף יכול לכסות את עצמו אבל לענין הפסק אפשר דמפסיק אע"ג דחד גופא אינון:
Bet Yosef makes a distinction between covering and separating, the same body may separate two areas (the one don't more see the other) but not to cover an area which needs to be covered because of Isur ervat davar. Baer Heytev reported that maharam Tiveli disagreed and said that a fortiori if covering is impossible, separating is also impossible. But Magen Avraham and Mishna Berura followed the Shulchan Aruch.
For Tachanun covering is needed, so the hand cannot cover.
The only Hava Amina to say that a non cloth object is not kisui is its non constant status. But concerning Tachanun the clothes don't cover constantly the face. So, no problem with non cloth covering. Additionally the problem is not really min bemino but "גוף אחד" "the same body "
Every external object is not "the same body" and can be used .
The reason for leaning forward when saying tachanun is that one should fall on one's face in shame for sins committed. The Arizal held the hand was not enough of a cover and one should use one's arm (not the tefillin arm). However one should not use one's bare flesh, as one's arm cannot sufficiently be a cover for one's face when leaning forward. In my siddur it mentions a sleeve or tallis but that assumes one is wearing a talis and one isn't wearing a short sleeve shirt. According to many, ideally one would be wearing a jacket during davening which would solve that problem but not everyone does. But then again neither does everyone hold like that Arizal that the hand is not enough of a cover for tachanun. Assuming that one is in the position of using one's forearm and having a short sleeve shirt it could be that one would be able to use anything to cover one's arm such as a paper towel. I don't think the watch would be a sufficient cover if the hand is not. While I haven't seen a source for it would there be any reason one couldn't rest one's forehead on an object such as a chair or edge of a table in this situation?