Right now it's traveling at about 0.55 miles per second. When I checked the other day that was over 0.7, and a bit after launch it was over 1.
Putting JWST into L2 is essentially the world's biggest curling toss ( https://en.m.wikipedia.org/wiki/Curling ).
As the OP alludes to, JWST's operational life is hard-limited by its fuel capacity. L2 (like L1 and L3, but unlike L4 and L5) is an unstable orbit, and without regular corrections it will drift out of position. This means that you want to save ALL your fuel for making those corrections, and thus spend as little of it as possible on getting there in the first place. So over the next month it's going to slowly crawl into place, cruising on no power other than what the Ariane initially gave it, getting continually slower and slower via the effect of the earth and sun's gravity until it stops perfectly in position.