'Dandelion' is, as you correctly pointed out, from the French, 'dent-de-lioun'.
early 15c., earlier dent-de-lioun (late 14c.), from Middle French dent de lion, literally "lion's tooth" (from its toothed leaves), translation of Medieval Latin dens leonis.
Other folk names, like tell-time refer to the custom of telling the time by blowing the white seed (the number of puffs required to blow them all off supposedly being the number of the hour), or to the plant's more authentic diuretic qualities, preserved in Middle English piss-a-bed and French pissenlit.
Looking into 'pissenlit' on a French etymology dictionary reveals:
BOT. Plante de la famille des Composées, vivace, à feuilles longues dentelées disposées en rosette, à fleurs jaunes et à petits fruits secs surmontés d'une aigrette. Synon. dent-de-lion (s.v. dent). Pissenlit officinal; feuille, fleur, graine, racine de pissenlit; salade de pissenlit; culture du pissenlit. Les feuilles radicales longues et couchées du pissenlit sont dentées, en lobes arqués et renversés, glabres et d'un beau vert. [...]
Comp. de pisse (forme du verbe pisser*), de la prép. en* et de lit*. Cette plante est ainsi nommée en raison de ses propriétés diurétiques. Fréq. abs. littér.: 77. Bbg. Schurter (H.). −Die Ausdrücke für den Löwenzahn im Gallo-romanischen. Halle, 1921 pp.8-24, 89-90, 105, 106, 110.
Which roughly translates to:
BOT. plant of the composite family, perennial, with long serrated leaves arranged in a rosette, yellow flowers and small dried fruit with a Pappus. Synon. Dandelion (s.v. tooth). Officinal dandelion; leaf, flower, seed, dandelion root; Dandelion salad; culture of the dandelion. The radical long, folded the dandelion leaves are toothed, lobes, arched and overturned, glabrous and green. [...]
Comp. piss (piss the verb form *), prep. in * and bed *. This plant is so named due to its diuretic properties. Freq. ABS. litter.: 77. BBG Schurter (H). −Die expressions as den Lowenzahn im Gallo-Roman. Halle, 1921 pp.8 - 24, 89-90, 105, 106, 110.
So apparently 'dent-de-lion' is synonymous with 'pissenlit', but didn't make mainstream usage, while England went the other way - apparently 'piss-a-bed' used to be another term for it, as Oxford shows:
Late 16th century: from the verb piss (because of its diuretic properties) + abed, suggested by the French name for the dandelion, pissenlit.
but we preferred to use 'dandelion' instead.1
Ultimately, both 'pissabed' and 'dandelion' are English words for the flower, and 'dent-de-lion' and 'pissenlit' are French for it as well, but the popularity seems to be swapped round, with English using dandelion and French using pissenlit.
This may be due to other European nations such as Italy using related etymology (pisacan).
1: Just speculating, but this could have been due to French-speaking nobility being in power, such as Henry VII and Henry VIII, and the French terms filtering down (like the difference between 'beef' and 'cow'). I'm not too sure about how language worked historically in France, and so can't offer any help as to why the French chose 'pissenlit' rather than 'dent-de-lion', possibly because of confusion with real lion's teeth?
editing in progress...