In Humfrey Barwick's 1594 discourse he briefly mentioned a longbow mounted on a tiller or a "tiller-bow" as being more accurate than a longbow alone and separate from a crossbow. Did full-sized infantry bows mounted on stocks ever see significant use in warfare or hunting in medieval Europe?

Interesting, I have never come across a tiller bow unless it was some kind of crossbow, essentially that is what a crossbow is, a bow set on a tiller. How was it supposed to differ from a crossbow?
By 1594 the longbow as a weapon of war was in a strong decline. A lot of the old texts are notoriously vague in their descriptions. I would have thought there would be no advantage in warfare adopting a tiller bow. Powerful metal bow crossbows, drawn with mechanical devices were already in existence. Heavy hand held longbows had a faster rate of fire, and a proven military track record. Armour was getting thicker with the ascendency of firearms.
As a wood bowyer for 20 years, wooden bows don't like being drawn for long periods of time. They take set, lose cast, and lose draw weight. All wood bows on crossbows were found to be inefficient, and composite sinew, horn bows were developed. They however took a long time to make, and could be negatively effected by wet weather, because of the solubility of the animal glues that held the sinew and horn together. Hence the use of steel bow crossbows.
Another issue with warfare in mind would be the lack of manoeuverability, with a six foot plus long bow, sideways on a tiller. Without a comparable powerful bow like the size needed for a wooden warbow it would not be a viable weapon against an armoured opponent.

Leo Todeschini might know more about this, if he could chime in here. He has made several different crossbow type weapons from the classical and medieval periods.

(It would be lovely if we had a 'tag someone' function here... ah well)