Just about everything is instant when I go Warrior, from Midas to Trials including mentor roulette which is solo queue only, and sometimes throws me in a Expert Roulette dungeon if it's not an ARR extreme primal that someone says they have been waiting a long time for. Slowest tank queue is always going to be Void Park, since it has the limited amount of roles but that is usually 10 minutes tops if solo and not full pre-form.
I would even say Expert Roulette isn't such a very high demand late in the cap week. As there are plenty of ways to cap with current content from repeat Midas runs, Seph Extreme, PvP, beast tribes, etc. Then you also have being able to fully preform for the other roulettes now like Trials, and we do that a lot. Mentor roulette tries to plug these holes though even if there is no one new to an instance. But that has been my experience as I try to do multiple times a day.