What is a longbow?

There isn't a universal strict definition of "longbow". Various archery groups have their own definitions, which vary. Some of these exclude modern materials, others exclude some old bows that are called "longbows" by most people.

A simple definition would be a straight wooden self-bow of approximately the same length as the archer's height. Some people would modify this to exclude long flatbows (with an arbitrary cut-off based on the ratio of limb thickness to width).

The earliest bows found (the Holmegaard bows) are longbows by the first of these definitions. Since their width:thickness ratios are a little over 2, they can easily be classified as flatbows. For measurements of the Holmegaard bows, see https://www.academia.edu/11765815/Mesolithic_Bows_from_Denmark_and_Northern_Europe

The oldest longbow excluding flatbows is that found with Ötzi the iceman, European Alps, buried in ice c. 3300BC. This bow was high draw weight. It has been suggested that the bow was an unfinished bowstave, a work in progress. This is based on the very high draw weight, and the wood being overstrained if drawn to 28". Measurements of the bow are available at http://www.primitiveways.com/Otzi's_bow.html along with a discussion of whether it was a finished bow or not. While it appears that the wood would be overstrained by a 28" draw, a 25"draw would only strain the wood as much as the bows it is compared with (1.3% strain), and the draw weight at 25" would be approximately 130lb. Thus, it is quite possibly a high draw weight longbow with a shortish draw length.

Also, was the Mongolian Bow not also a version of a longbow?

By essentially all definitions of a longbow, not, the Mongol bow isn't a longbow. First, it is short (that is, the Mongol conqest-era bow is short; recent Mongolian bows can be quite large, influenced by the Manchu/Qing bow). Second, it is a recurve-reflex composite bow rather than a straight self-bow.

However, if "longbow" is used to mean "high draw weight bow" (or "warbow", as many would say), it is certainly that. For some recent discussion of draw weight of Mongol bows, see https://www.reddit.com/r/AskHistorians/comments/928g79/it_is_said_mongol_bows_draw_weight_was_as_high_as/ (in summary, they could be over 150lb, but probably usually somewhat lower).

what took the Longbow so long to come around? Why did the British have it, but not the French?

The problem isn't inventing such a bow. Longbows are very old bows, and quite possibly very high draw weight longbows are very old. High draw weight longbows have been independently developed in multiple parts of the world, so it isn't a technological problem.

For very high draw weight longbows to become major military weapons, there are some factors that need to be considered:

1. Need: Are very high draw weight bows needed? Very high draw weights provide arrows with lots of kinetic energy, which is the key to shooting through armour. In the absence of armour, or if armour is rare, very high draw weights aren't needed. Unless they are necessary, they are best avoided - demanding high draw weights reduced the number of archers available who can use the bows, and can limit the effectiveness of archers if they are underfed, ill, or exhausted (all of which soldiers often are).

2. Need part 2: Is a very high draw weight bow other than a longbow available, and sufficiently cheap? If yes, there is no need to use a longbow (one might use, e.g., the Central Asian composite bow, such as the Mongol bow).

3. Are archers available in sufficient numbers? If insufficient trained archers who can effectively use such bows are available, they might appear on the battlefield but in too small numbers to matter. High draw weight longbows were available to the French, but their efforts to develop sufficient numbers of skilled archers failed.

4. To be effective in battle, archers need to be supported by other soldiers. English longbowmen failed in battle on many occasions. Usually, without field fortifications or protection from other soldiers (e.g., armoured infantry to stop attacking enemy soldiers), they were unable to stop attacking cavalry. It is important to have effective combined arms tactics in order to make effective use of massed archers.

Given armour, draw weights will tend to climb to where they challenge the armour. Armour tends to keep up, and provide good protection, at least as far as torso and head armour goes. Generally, limb armour is lighter, and will not reliably stop arrows at short range from high-power bows. But apart from the bows, one needs archers in sufficient numbers (3 above) and the tactics to make use of them (4 above) before they can be a key weapon on the battlefield.

the Longbow was OP, with an enormous draw weight and range and the ability to pierce most armour like leather, gambeson, and mail.

According to the tests described in Alan R. Williams, The Knight and the Blast Furnace, Brill, 2003, 120J is needed for an arrow to defeat mail and the backing textile armour. If the armour is a sandwich of textile-mail-textile, with an additional layer of textile armour on top of the mail (as was used historically, at times), even more energy would be needed. A high draw weight longbow can provide 120J, or more (but not much more) at short ranges. Mail is not easily defeated by archery, and heavier stronger mail, and mail with textile armour over the top, may well stop the most powerful longbows even at very short range.

The longbow doesn't win battles through being OP, by going through armour as if it wasn't there. It wins battles by causing some casualties (some arrows go through armour, and some arrows hit limbs which are often much more lightly armour than the torso), by killing horses (or forcing the enemy to attack dismounted), and by continuing to inflict casualties on enemy engaged with one's own armoured infantry (archers can shoot at such enemy at short range in relative safety for a while). If a large part of the enemy force is unarmoured, and has restricted mobility due to threat of cavalry attack, archery can be very effective. If the archers aren't protected from enemy attack, defeat can result (as the English learned against both the Scots and the French). In summary, effective, but not OP.

This is going to be a long answer but here goes.

Although there is no accepted standard classification system for bows, it is generally agreed that a longbow has four primary characteristics: it is longer than a ‘regular’ bow (usually around about the height of the archer if not a little longer), it is a ‘self bow’ (that is, it is made from a single piece of wood), it is not recurved and it has a circular or D-shaped cross section.

The vast majority of traditional European bows were self-bows, but not all were longbows. Generally they were constructed in the same manner and often the same materials as a longbow, but were simply not as long and thus had a shorter draw. The wood of choice for a longbow has always been yew, because of its great compressive strength, light weight and elasticity. However, other woods, such as elm, are known to have been used in the absence of yew. While a longbow consists of a single yew stave, it actually contains two types of wood. The outer ‘back’ of the stave consists of the yew tree’s sapwood, while the inner ‘belly’ is made from the heartwood. The former is younger wood and is more elastic, while the latter is older wood and more resistant to compression. This combination gives the longbow a heavy draw weight (because the stave is more resistant to being bent under tension by the archer. Altering the sapwood/heartwood ratio can produce a heavier or lighter draw weight as desired) and thus makes it more powerful. Combine this power with a full draw ‘to the ear’ by the archer (made possible by the longer length of the stave) and you’ve got yourself an incredibly potent weapon. Add a heavy war arrow and you’ve got a weapon capable of piercing most armour at close enough range (the exception being plate armour, though the thinner armour on the limbs was likely more vulnerable than the thick breastplate)

Of course longbows are intrinsically associated with England and, to lesser extent, Wales. However, in reality longbows have been used by many cultures for a very, very long time. Indeed, the oldest known longbow was discovered with the remains of Otzi the Iceman, who died in Alps around 3,300 BCE. His bow was 1.82m tall and made from a single stave of yew, just like the famous English warbow. Ancient longbows have also been found in peat bogs in England (dated to around 2,600 BCE) and in Denmark (a cache of 40 specimens dated to the 4th Century CE).

Now we get to the English question. Why did the English have longbows and the French didn’t? The simply answer to the question you have asked is that in truth the French did have longbows, though possibly not quite as powerful as the so-called ‘English longbow’, which was a heavy bow designed specifically for war (as opposed to hunting). However that is still a subject of significant debate and unfortunately no specimens from the height of English dominance have survived (there are many Renaissance longbows that have survived but there is debate about whether or not they were the same/as powerful as those used in the 13th-15th centuries). Anyway, the question I believe you were trying to ask is why did the French not field their own longbowmen at Crecy, Poitiers and Agincourt? The answer to this is that it was not a matter of an English technological advantage, but rather it was simply a matter of scale. By the 13th century, English kings had long seen the potential of the heavy longbow in war. Initially they had employed Welsh longbowmen, but eventually they began training their own. This was not just a matter of making longbows (which traditionally takes 1-2 years a piece) and giving them to potential archers. It takes years and years of training and practice to be able to effectively wield a longbow. This represents the kind of investment that can’t really be made on a campaign-to-campaign basis. In order to create a huge pool of ready-made longbowmen, Henry III ordered that all yeomen must be trained in the use of the longbow. This was later reinforced and expanded by Edward III to include all able-bodied male villagers. As a result, village children began training (famously using rocks as weights) at a young age to build their strength. Edward began the process of embedding the use of the longbow into English society. To arm this martial class, he oversaw the development of a massive manufacturing network, which produced thousands of longbows and hundreds of thousands of arrows every year. This was the reason the English were able to field thousands of longbowmen at Crécy, Poitiers and Agincourt and this was the reason the French and others could not do the same. They simply did not have the systems necessary to produce the archer, the bow or the arrows that were required on a large scale.

As to the last question, no, Mongolian bows were/are not longbows. They are recurved and they are composite bows (ie constructed by laminating layers of horn, sinew and wood together, thereby creating a similar effect as the sapwood/heartwood combination) rather than self bows. They are also much shorter. However, for all that, they are still very powerful bows.

Hope this long reply gives you the answers you seek.