The earliest mention of a Coat of Plates or something like it is in an early 13th century account of a combat of Richard I of England in the late 12th century - this could be an anacrhonism, but since the chronicler died in 1225 it is from the first quarter of the 13th century at the latest. Mentions in accounts start in earnest after 1250 and there are two depictions in art dating from the second half of the 13th century. By the end of the century some inventories list thousands of them (this is as per a translation from Williams). Some effigies in the early 14th century show no visible plate defenses but it is unclear if a Coat of Plates was worn under the surcoat. By 1330 they seem nearly universal. In the 13th century they took the form of similarly sized oblong Plates oriented vertically. By the middle of the 14th century they sometimes included more specially shaped Plates, as we see in the Battle of Visby finds (which are thought to have mostly been older designs).

Their replacement by the full plate cuirass is harder to define, because over time different parts are replaced by bigger and bigger solid pieces of metal. In the middle of the 14th century the breastplate is sometimes one solid piiece, and over the next 25 years or so the pieces below the breastplate form a fauld or skirt of hoops. Sometimes the breastplate us in two pieces and is joined at the middle by buckles (like a later brigandine). The backplate is slower to develop as one piece of metal, as is the culet, the skirt covering the rump. The backplate is sometimes made of multiple pieces and sometimes absent. Moreover, as a general rule in all this time period these elements were generally still covered by cloth. Also, the term brigandine is used in this time. Around 1400 or so the breast and backplate are more frequently worn together and without a cover. This is the end of the Coat of Plates but covered armour lives on in Brigandines. It is important to note that there is no clear single progression and that many breastplate and coats of Plates coexisted at once. Moreover, old forms of armour could still be used, as was the case with many of the Visby armours.

So what you have is the 'classic' Coat of Plates having a heyday of some time between 1250 and 1350 and transitioning into more and more breastplate-like forms between 1350 and 1400, and in the 15th century we have brigandines.

For the Cuirass of cuir brulee, it appears to be a supplemental armour used between the 13th and the first half of the 14th century.

EDIT: Blair mentions it in accounts in the second half of the 12th century.

Sources: Claude Blair - European Armour c1066-c1700

Alan Williams - The Knight and the Blast Furnace

Edge and Paddock - Arms and Armour ofcthe Medieval Knight.