Although we accept the opinion among the Rishonim that it is forbidden on Shabbat to reheat boiled water that has cooled down,[1] in the case you describe, you would not be considered to be cooking, since this is done automatically. The question is whether your setup violates the Rabbinical prohibitions of shehiya or chazara.
Shehiya, leaving food on the “flame”[2] from before Shabbat, is sometimes forbidden, out of a concern that one will raise the flame on Shabbat. It is permitted if the heat source is covered in a way that reduces its efficiency[3] or (likely) regarding a nonadjustable heat source.[4] However, neither lenient factor exists in your case; you can raise the heat of the urn by switching from the Shabbat setting to the normal mode. A common claim for leniency in the case of a water urn is that once the water has already been boiled, further boiling causes unwanted evaporation, and we therefore are not concerned that one will raise the heat.[5] However, your case could possibly be more problematic since the Shabbat timer turns the urn on at low heat, and one may desire extra heat to return the cooled-off water to the desired temperature more quickly.
Chazara, returning on Shabbat food that had been removed from the heat, is treated with more stringency. For example, it is forbidden to do chazara on an uncovered, adjustable heat source even when raising the temperature is detrimental to the food.[6] Is your case considered chazara, given that the heat source starts functioning again through the action of a machine, rather than a person? The answer may depend on the reason for the stringency of chazara in comparison to shehiya. Rabbeinu Tam maintains that it is due to a heightened concern that one will raise the heat since the food was returned after being off the flame for a while.[7] However, the Ran[8] maintains that the added problem is that returning cooked food to a heat source may be confused with cooking. In your case, Rabbeinu Tam’s reason seems to apply, as you may be inclined to raise the heat of the urn; the Ran’s reason does not apply, since you are doing no action on Shabbat.
Let us examine a discussion about a parallel case. The Rama[9] writes that on a winter Shabbat morning, a non-Jew may put cold cooked food near a fireplace, which a non-Jew is permitted to light for a Jew due to the extreme cold, thereby also heating the food. Why aren’t we concerned that a Jew will stoke the burning coals? The Pri Megadim[10] suggests that this ruling must rely on the opinion that reheating liquids is permitted. Consequently, the reheating is not significant enough to forbid due to the concern that one might stoke the coals. The Chazon Ish[11] suggests other possible answers. One is that if a food is put in a position with intention to heat it but there is presently no heat, we treat the situation as equivalent to shehiya. Since the Chazon Ish claims elsewhere[12] that the concern of raising the flame regarding shehiya does not apply to fully cooked food, even if it is now cold, this explains the Rama’s ruling.
Despite the fact that this view of the Chazon Ish would seem to permit reheating in the situation that you describe, reheating cooled water may be worse than reheating other cooked foods.[13] Although there is room for leniency given that the urn was operating when Shabbat began and no action was taken since then,[14] the Shemirat Shabbat K’Hilchata[15] is stringent when the water has cooled off totally, and the Orchot Shabbat[16] is uncertain.
However, your urn has a feature that provides further support for leniency: When the Shabbat mode is on, one cannot raise the heat. This is similar, in some ways, to one who seals an oven that has burning coals inside where food is heating, which is permitted even though the seal can be removed.[17] It is unclear whether the Shabbat-mode button, which is deactivated by a simple press, is a sufficient deterrent,[18] and it is also unclear whether this leniency applies when elements of chazara exist.[19] However, combining this factor along with the aforementioned grounds for leniency, it is not difficult to justify your practice.