Well, chariots were in fact used by the Celtic tribes of Gaul and Britain up to the first century AD. In fact, the English word chariot comes from the French charriote, which itself comes from the Latin carrum, which is a loan word from the Gaulish karros. The Celtic chariot would have been constructed of wood and wicker, with the wheels rimmed in iron and iron wheel hubs. Each chariot would carry two men, one driver and one fighter. Julius Caesar wrote a first hand account of the method British Celts used while fighting with their chariots during his invasion of Britain in De Bello Gallico

Their mode of fighting with their chariots is this: firstly, they drive about in all directions and throw their weapons and generally break the ranks of the enemy with the very dread of their horses and the noise of their wheels; and when they have worked themselves in between the troops of horse, leap from their chariots and engage on foot. The charioteers in the meantime withdraw some little distance from the battle, and so place themselves with the chariots that, if their masters are overpowered by the number of the enemy, they may have a ready retreat to their own troops. Thus they display in battle the speed of horse, [together with] the firmness of infantry; and by daily practice and exercise attain to such expertness that they are accustomed, even on a declining and steep place, to check their horses at full speed, and manage and turn them in an instant and run along the pole, and stand on the yoke, and thence betake themselves with the greatest celerity to their chariots again

Also, Queen Boudicca famously rode a chariot into battle against the Romans in her revolt in 61 AD. Tacitus describes her as riding her chariot from tribe to tribe, begging for assistance in fighting the romans, and extorting the other tribes to rebel. She then commanded her army during the Battle of Watling Street from her chariot.

Where chariots ever used by European powers?

As /u/Danegeld87 mentions, chariots were used extensively by the Celts.

If not, when and why did the chariot stop being used in combat?

Chariots have many disadvantages which are likely reasons as to why their importance on the battlefield declined into irrelevance by the height of Classical Antiquity.

1. Chariots are more difficult to control than individual horses. Scythed chariots in particular had a tendency of slicing up the legs of friendly infantry. Cavalry can pull complex maneuvers that chariots simply can't, at least not easily.

2. If one of the horses pulling a chariot is incapacitated, the chariot becomes useless. As the horses are big targets for infantry (especially archers), this is a significant vulnerability.

3. Cavalry is a more logically efficient use of horses. Most war chariots had to be pulled by more than one horse. As a result, an army can field more cavalry than chariots with the same amount of horses, or keep horses in reserve as spare mounts.

4. Horses are very good at crossing many type of terrain. Chariots are not, and chariots generally require quite a lot of open land to be used in battle effectively.

Chariots were once the dominant "mobile" arm in many ancient armies because the breeds of horses at the time just weren't strong enough to be effective cavalry mounts. As selective breeding led to stronger horses, cavalry begins to supplant chariots.

Chariots did remain in some use well into the classical era, though. The Celts used them, as aforementioned, quite effectively. Scythed chariots were employed by the Achaemenid Persian Empire against Alexander the Great, and the Kingdom of Pontus used them against the Romans. The Achaemenids and Pontus came out on the losing side of their respective wars, however, which furthered the decline of the devices, though some of the Hellenistic kingdoms did use them now and then. Scythed chariots generally tended to simply be more trouble than they were worth, and only effective in very specific conditions. By the time the Hellenistic Kingdoms fell to Rome and the Parthians moved in to dominate Persia, chariots are pretty much done in that part of the world.

I'm no expert on Chinese history, but chariots seem to have faded out of widespread use by the time of the Han dynasty (206 BC–220 AD). The Han did use armoured chariots against the nomadic Xiongnu by essentially using them to shield their archers, but based on that description I suspect that the chariots being used were really more similar to war wagons than to normal chariots.

The Celts in Britain and Gaul used them against the Romans, but the Celts in Britain and Gaul also ended up losing out to the Romans, who themselves weren't too keen on chariot warfare. They might have remained in use in Scotland and Ireland, beyond Roman control, but they seem to have eventually died out even there.

Chariots can be effective in battle, but cavalry is just much more versatile. Chariots would also be highly vulnerable to archers, since the horses are quite vulnerable. I suspect that the prevalence of crossbows in China had something to do with their eventual disappearance in that part of the world. And the last few groups of people who did use chariots had a tendency to be conquered by those who didn't use chariots.

So just to clarify, when you say European powers, you're talking about Europe after the Western Roman empire? Like, say, after the 5th century?