Artillery had been in use within Europe for over a century prior to the War of the Roses, which meant that most armies during this period of viloence had some form of artillery.
This wikipedia page gives a good overview of artillery in the medieval period, which is a good start in researching the subject.
Artillery pieces during this period were considerably slow to load and fire. Projectile were either made from stone or iron that could weigh up to several hundred pounds, which made the process of loading very slow. To fire a wepon of this period gunners used a firing iron—an iron bar heated in a pan of charcoal that was kept hot and near at hand. Each shot generally required 1lb of powder for every 9lbs of shot, once fired the barfrel would need cleaning after every round, which further slowed the whole process. A good gun crew could be expected to fire a rate of 10 shots an hour. Due to this slow rate of fire, cannon were generally used on the eve/start of battle firing 1 or 2 volleys, before troops engaged in the melee of close combat.
Artillery played a lesser role during the War of the Roses, compared to conflicts on the continet, with the ability to fire a projectile 2,000 to 2,500 paces, they could be used with dramatic results against massed troops who were imobile, such those at a river crossing. Sieges were not a common occurance in Eglish conflicts, as fortifications were less sophiticated than thier European counterparts, and the War of the Roses pitched battles were more common place. Most towns, castles would more likely to surrender once its army had been defeated on the field of battle than attempt to withstand a siege.
The Wars of the Roses by Alison Weir, Ballantine Books.
Medieval Siege Weapons: Western Europe Pt. by David Nicolle (Author), Osprey Publishing
Mediaeval Siege Warfare by Christopher Gravett, Osprey Publishing
Ancient and Medieval Siege Weapons: A Fully Illustrated Guide to Siege Weapons and Tactics by Konstantin S. Nossov, The Lyons Press