When does each class get "efficient"?

[Cyrylius]

So, regarding the runepriest/WP comparison: runepriest, as a dedicated single target focus healer, is possibly the first (alongside mirror zealot) class to reach its full potential, as your actual class defining skills in ST flash heal and bubble come in first 10 levels. On the flip side, you don't get anything else for the whole time you are playing the class (barring maybe winds of insanity later on the mastery tree and group heals, one of them you still get very early anyway).
WP/ DoK, on the other hand, are tankier (that doesn't change much initially, as the difference is too small and incomparable to the mobility advantage of zealots) and slower, while also having a bit more group healing potential. You can also play them in three different specs, all of them being viable in some cases. While DPS spec is probably going to be underwhelming until late into the game, at about rank 20 you can play the book/chalice or snb better than heal zealots or runies in terms of sheer healing numbers (worth mentioning: that's not everything that matters about healing).

About other classes: it heavily depends. Sorcerer and bright wizard are good in T1 for reasons already mentioned, and they get the ST toolkit reasonably fast too. Magus/engi are great as they do magic damage that will shred everything until late 30s.
Marauder and white lion differ a bit, as marauder gets heal debuff very fast-albeit only the weaker one. WL on the other hand doesn't get the trademark burst before t2, where you get coordinated strike (don't quote me on that, might be late T1). Also, marauder gets spammable AoE early on, while lion will get its one only in late t2. Tanks in general don't get to do any tanking before getting the guard and challenge, though worth mentioning that BG is the only one getting punt early, and it's the best punt in game to boot which makes its initial levels much more engaging. AM/shammy will by about rank 10 get enough skills that you can play around with the class mechanic to great success. Choppa/slayer will be pretty similar throughout the leveling process, both getting aoe skills almost instantly and not needing any specific abilities to function properly, except maybe the heal debuff. Finally, I have no idea about squig herder and shadow warrior.

Summarising, most classes will have their core toolkit by rank 20, and some will get it faster. Then some classes need the specific mastery abilities or tactics, which means at around rank 30 all classes should play similarly to how they will be played in the endgame (WP/dok aoe channel, marauder/WL AOE channel/spammable, 2h BG crit debuff, to name a few) and all the differences that come afterwards are usually comparatively marginal. Finally. I'm not talking about off specs, which usually require specific, often high level, gear, and lots of mastery points.

Edit: we/wh will seem pretty strong from the beginning as they get heal debuff already in T1, however you don't get the stealth until level 11 which makes playing pretty awkward prior to that.