It can be a little confusing because Blizzard uses Timewalking as the name for two different things.
You have Timewalking Campaigns (usually called Chromie Time by the community) which lets you level up to 60 using any expansion.
You also have Timewalking event weeks, which is what Laicoris was referring to:
These occur every few weeks and cycle through all of the expansions that have been added to the events. (Right now, Legion is the latest one in the pool.) The Warlords of Draenor timewalking event is active this week. If you look at your character’s buffs, you’ll see one of them is active for Draenor Timewalking.
During a Timewaking week, the dungeon finder will have the option to queue for a random Timewalking dungeon which puts you in one of the dungeons from the current event’s expansion. You are scaled down to the max level for that expansion while in the dungeon, but the gear that drops will be scaled to your current level. You need to be at the natural cap (or higher) for the expansion to access the dungeons.
BC + Wrath: 30
Cata + MoP: 35
WoD: 40
Legion: 45
You can earn a currency, Timewarped Badges. You get 5 off of bosses, 10 for completing a dungeon, and when you kill the last boss of your first Timewalking dungeon during the event, you’ll get a quest for 500 badges (that’s the big source of them each time it’s up). Each Timewalking event has their own vendor that is up during the event (the 500 badge quest leads you to them) that sells stuff like mounts, pets, toys, and as Laicoris mentioned, rep tokens for the various factions of the expansion.
Rep gains are also increased with factions of that expansion during the event.
The Legion event is scheduled for the beginning of April, so it’s not that far off. (I’d still just figure out where you left off with questing when you got kicked out of Chromie Time— it will be way faster than waiting, and at this point I’d save the Timewarped badges for the fun stuff.)