I happen to be doing a bit of research on this at the moment, looking at gunpowder weapons in 14th century London. I can't really comment on explosives, because England doesn't seem to have bothered with them, but I can talk a lot about cannons.

Short Answer

Simple gunpowder weapons emerged in Europe in the 1320s. The first effective weapons emerged around 1340 and they became common, but they were small. The first wall-smashing artillery pieces were created in the 1380s. By the early 15th century, artillery pieces were a common part of any decent army.

Long Answer

The recipe for gunpowder seems to have been in circulation from the late 13th century. Exactly how Europeans became aware of gunpowder and gunpowder weapons is a topic of some debate, and it was very likely a combination of different avenues. By the 1320s the first gunpowder weapons saw action on the battlefield, and some are described in two London manuscripts, De Nobilitatibus, sapientii et prudentiis regum, a book of the ‘mirror for princes’ genre which contained a section on gunpowder warfare. And another called De secretis secretorum Aristotelis. Both were produced in 1326 and presented to king Edward III at his coronation.

These weapons were really terrible. The main type was the pot-de-fer, a vase-shaped cannon that fired an arrow which was plugged into it with some wadding, a bit like a champagne cork. Reconstructions have found that it was rubbish, accounts we have generally give no tactical significance to it other than being a novel technology, but it was something, and enough people saw the potential in the weapon to develop it further. Smaller versions became available and they started to be loaded with lead shot as well as arrows/bolts by 1340. They were more dangerous, but only at short range and their accuracy was awful.

Then in 1337 the Hundred Years War began, which was arguably the main catalyst for the development of black powder weapons in western Europe. France already had some black powder weapons, but Edward III seems to have had 0. The ranged weapons of the king's personal forces were managed by the Royal Wardrobe and the Tower of London, and we have the financial records for them. They had no guns. In fact, the only black powder related activity they did was the purchase of small quantities of raw ingredients for an unknown purpose. The English army did have at least one cannon, mentioned by the French Chronicle of London in the 1341 entry, but that's it. However, around the same time a new type of weapon called a ribault (or rebaultkin, rebilt, rebaltkine etc. the spelling of it is all over the place) emerged which was a series of gun barrels set up on a cart. The operator could quickly fire off a volley of a dozen shots into the enemy. English forces faced some of these guns in the late 1330s and early 1340s, and lost consistently. This seems to have impressed the potential value of black powder weapons upon the English king.

When Edward III prepared to launch a major attack on Normandy, planned for 1346, he ordered the repair of captured guns and the production of new ones. He ordered the Tower of London to make 100 ribaults with only a few months of notice, and only two people were tasked to do it (though they were given about £124, presumably to hire additional labour from London). Only 5 of those guns were actually ready in time, but they performed well at the Battle of Crecy. The Battle of Crecy is usually discussed in relation to English longbow tactics, but it is also a watershed moment in the development of artillery in Europe. From then on, the nations involved in the Hundred Years War tried to develop better and better artillery to outdo one another. Gunpowder was a regular feature of the battlefield from this point in the war.

These cannons were not big. London manufacturers had the great idea of charging the crown by the weight of the cannons, so we know how heavy they were. Cannons from 1340-60ish are rarely more than 200lbs (90kg). From 1380-1400, the 'standard' cannon produced by London's foundry was only 380lbs (172kg), with 47 being made from 1382-8. The largest cannon was 1590lbs (720kg), and only 5 were made. This latter type of cannon, usually called a 'bombard', was designed to smash castle walls. Whomever built a cannon capable of smashing down a castle wall obviously had a huge advantage, so there was a race to develop bigger cannons. France, England, and the increasingly involved power of Burgundy poured money into developing these cannons and centralised their research efforts. The Tower of London records from 1388 show that almost £1800 was spent on black powder weapons for the king's personal arsenal. References to gunpowder weapons in contemporary sources skyrocket at this time, they were everywhere.

In the 15th century the technology continued to accelerate. In the mid-15th century Burgundy was able to produce bombards such as the Mons Meg, which weighs 13,000lbs and fires a 175kg shot. Flemmish gunners became a regular part of English armies, and gunnery squadrons began to appear so often that they were no longer worth mentioning by contemporaries.