How common was banditry in 16th & 17th century Great Britain and Ireland?

There are quite a few great sources on this topic. If by “how common”, you are implying that you are looking for hard, measurable and very-much-incomplete sample data (that you have to, of course, collate yourself), this is going to come from digitized historical court records like the Assize Courts that @Kobunite linked to, or from the proceedings of the Old Bailey. If you’re looking for more comprehensive discussions and original, primary-source commentary on the phenomenon of highway banditry in the British Isles during the Tudor/Stuart eras, these remain in that ancient, inaccessible format known as "books"... (I'll cite them throughout this post).

Perception vs. Actual Incidence

I think the best place to start is to acknowledge that the modern cultures of Britain and Ireland both contain prominent mythologies about famous hero-bandits and their respective subcultures of highway banditry (these have in turn birthed American and Australian robber-hero narratives that you're probably familiar with). In Outlaws & Highwaymen, Spraggs references several contemporary sources that make it clear that banditry was consistently PERCEIVED to be very common in Great Britain and Ireland, all the way from the beginning of the Hundred Years’ War to the First Industrial Revolution.

To over-generalize so that I don't write an entire paper here, these contemporary commentaries throughout the centuries consistently fall into one of two categories: they either venerate the culture of mounted banditry as a "gentleman's sport" (and later in Stuart England/Ireland as acts of political defiance) or describe it as a serious social problem. The extent to which they are based on factually recorded instances of banditry vs. inflated perceptions makes for a great debate in the faculty break room of a college humanities department. And I think it's instructive that although we have insane amounts of measurable, accessible data about our own society today, common perceptions are usually way off base (For example, Gallup and USA Today have released surveys in the past few years that say a majority Americans believe crime rates keep rising, even though they've fallen consistently for the past quarter century). The point is that this topic is heavily-reliant on primary-source COMMENTARIES rather than primary-source DATA. Hence, any surface-level research like what's presented in this answer is going to give you a historically accurate summary of 16th & 17th century British & Irish cultural understandings of contemporary banditry, and not an objective analysis of records of specific instances of banditry.

Two Hundred Years'-Worth of Highway Robbery

From an events-based standpoint, I think it's also worth pointing out that your question is probably broader than you may think. The way it’s currently framed (Highway Bandits x (England + Ireland + 16th & 17th Centuries) - Smugglers & Urban Criminals = ???) means that a thorough answer needs to be broken down into multiple (closely-related) categories. Why? Because the historical record shows us that the motivations for (and contemporary perceptions of) banditry in England in this time period vs Ireland were actually somewhat divergent, owing to the unique political situations and cultural traditions of banditry in those respective countries. Additionally the 200-year span that you've defined captures two very distinct epochs of Anglo-Irish history, which can be broadly defined as the Tudor/Elizabethan period, and the reign of the House of Stuart (ending of course, with the Glorious Revolution in 1688). Instances of banditry in these periods and places weren't all calculated acts of economic desperation as one might expect. Many of them were just as much deliberate acts of political, sectarian, and even sporting violence. Thus, the highway robbers circuiting the outskirts of Elizabethan London were a different class of men than those waylaying sheepherders in the Cromwellian Midlands, who were in turn, different from the Irish “tories” harassing English settlers in Cork and Scottish Presbyterians in Tyrone during this time period.

Banditry's origins

All the way back in 1516, we can find Thomas More lamenting in Utopia that “theves were in every place so ryffe and so rancke”, here referring to England’s curious subculture of highway banditry and reflecting that it is a sad product of the economic turmoil that has afflicted England ever since the Tudor accession. Yet if we look farther back in English history, we find that the motivations for banditry were often not purely economic, and the bandits themselves did not uniformly hail from the poorest classes of society (which one could understandably assume). [See also, A.V. Judges’ The Elizabethan Underworld]. In fact, quite the opposite is true.

In England, the key tools of highway banditry have always been a functionally lethal weapon (a sword or a pistol) and a means of a quick getaway (a horse). Until the 18th century, neither of these items were easily obtainable for England’s poorest classes (i.e. the people who would likely resort to highway robbery out of economic necessity). Though “footpads”--bands of peasant-thieves roving the countryside and engaging in what you might call forage-thieving, stealing chickens, foodstuffs, farm tools, and horses if they could find them--weren’t uncommon, there is evidence that these types of bandits were as often-as-not seasonally-unemployed laborers as they were career criminals. Making a living by robbing travelers on the road took a set of resources that were out of a peasant thief’s reach.

John Bellamy in Crime and Public Order in Medieval England documents several court cases and written eye-witness accounts from the 14th century where knights (as in well-heeled members of the land-holding warrior class, those kinds of knights) were observed threatening traveling farmers and merchants at the point of a lance and making off with their valuable goods and/or money. Spraggs dubs them “gentlemen gangsters”. His analysis concludes that this kind of activity was viewed by its perpetrators as almost sporting in nature. A well-paraphrased translation spells it out: “Thank God” an anonymous Franciscan friar remarked sarcastically, “that we live in a country without robbers, only robber barons.” Keep in mind that the tale of Robin Hood, a folk narrative that turns this exact trope on its head, was invented around this time...

Tudor England

Fast-forward to the time of the Tudors. The “gentleman gangster” mentality appears to have infected the minds of the young, restless heirs of London’s wealthy merchant class, and there is a good deal of primary source evidence to support this claim. For a couple examples, we have William Harrison’s Description of England (1587), in which he observes that the bandits roving the highways to London are “extravagant young gentlemen”. And we have the Jesuit minister Robert Parsons, who in 1596, observed that so many of the men committing robbery on horseback with pistol in hand were “gentlemen’s sons”. (Again, this makes sense because horses and pistols weren’t cheap). Even though the legal penalty for this type of crime in Elizabethan England was death by hanging, Parsons’ complaint was that it was rarely enforced because so many with means engaged in it. Both Shakespeare (King Henry IV, Part One) and Ben Johnson (Every Man Out of His Humour) thought that the cultural glorification of the highwayman's lifestyle, especially among cohorts of well-heeled urban youth, was a serious enough problem in their time to merit a thorough skewering in their respective plays.

Bear in mind that no argument is being made that even a significant minority of the instances of highway robberies in Tudor England were committed by men from the knighted, noble or landed classes (we don’t have that kind of data). But we can confirm from primary source commentaries and court records that it was a consistent, observable phenomenon going back to the Middle Ages.

Stuart England

This is the time period when we see an explosion of letters, pamphlets, and compendiums detailing (and in many cases celebrating) England’s historical culture of banditry. In 1617, the word “highwayman” enters the English language, and not long after, the notion of the mounted highway robbery as a “gentlemanly crime” crystalizes in the English public mind. John Clavell, James Hind, and Claude du Vall are all gentleman robbers who attain celebrity status for their exploits, though Hind and du Vall are both eventually arrested and executed.

Speaking of Hind, it’s important to note that he was an outspoken Royalist (aka Cavalier, aka supporter of King Charles against Oliver Cromwell). When he was arrested by Cromwell’s government, he confessed freely that he was a highway robber, arguing that he had committed no crimes as he had only waylaid and murdered Parliamentarians, who in his eyes were traitors to the King. Hind is a good example of how the phenomenon of gentlemanly criminality (as seen in the the actions of the Cavalier class) was applied to political ends during the English Civil War. In the Elizabethan era, an outlaw like Hind might have been considered "noble" by some, but still an outlaw. It was only through the cultural churn and upheaval of the English Civil War that he and outlaws like him morphed into irregular fighter for losing political cause. If you would like to read more on this interesting subtopic, I’d suggest checking out John Barratt’s Cavaliers: The Royalist Army at War.

16th/17th Century Ireland

Still with me? Let’s talk about Ireland for a second. Stephen Dunford’s The Irish Highwaymen is a good catalog of Ireland’s narrative tradition of the romantic highway robber (hint: the robber-as-hero trope isn't unique to England). We have less data on the actual number of recorded acts of banditry, but the records we do have portray this mischief as acts of explicit political resistance to British occupation, so basically skirting the definitions of guerilla warfare. The most famous political bandit from the 17th century is Redmond O’Hanlon, sometimes called the “Irish Robin Hood”. The word tory (Gaelic tórai, meaning "raider") was first used politically to refer to O'Hanlon and his fellow bandit-guerillas. (It was later retroactively applied to England’s Cavalier class, and only later became the nickname of members of Britain’s Conservative party.) The evolution of the meaning of this word is a fascinating subject in and of itself; it pretty much sums up the whole gangster-turned-gentleman theme that's emerged from this post. In modern Irish history and folklore, you can find a litany of other political hero-bandits. When they were caught, it was usually by British-backed authorities. In many cases, they were not charged, tried and executed for the criminal acts of theft or murder, but rather for challenging the sovereignty of the British Crown.

Again, this is NOT a representative criminology survey of highway banditry in 16th & 17th Century Britain and Ireland, nor an analysis of the economic or social constructors that made it possible. It is just a narrowly-defined portrait of its contemporary perceptions and cultural functions.