England is an interesting example. The short answer is that for fully armoured men at arms (knights and those who fought as knights) shields were not used in war in the 15th century. However some shields were used by less fully armoured infantry. In England, this particularly means archers, who carried bucklers and swords to defend themselves when armies closed (archers were close combat troops as well as ranged troops). I speak about the weapons of the the later 15th century more here and weapons of the earlier 15th century here. I haven't studied Tudor warfare in as much detail, however I will say that in general fully armoured men at arms did not use shields in that period either. Fully armoured men at arms -would- use single-handed swords as side-arms - swords are extremely versatile, and unlike polearms can be carried along with other weapons.

However, since we are speaking about soldiers fighting on foot, specifically, I think it is important to note that even earlier, in the 14th century, we see English men at arms using two-handed weapons rather than swords and shields. In particular, English men at arms often carried 'short lances' which would be wielded as spears, generally in two hands; while the inventories of the Tower armouries from the 14th century have hundreds of long lances, they have only a handful of 'scuti' or knightly shields (they have more 'targets' - smaller shields for other infantryman, and pavises, used the protect crossbowmen and other infantry from missile weapons). So even in the 14th century two-handed weapons were already the dominant weapons of heavily armoured infantry. Shields were used by more lightly armoured infantry.