Supposedly, Kaladin bonded Syl (at least, first interacted with her) just before the battle when he killed Helaran. By that time he was already an accomplished soldier and squadleader.
I think in one of Teft's POV he hypothesised that stormlight and the Nahel bond doesn't grant skill, it perfects. Probably Hauka already possess some amount of capacity in that direction. I don't think Kaladin consciously Invests before speaking the First Ideal, but he does use surges: he displays unusual amount of strength, endurance, deflect s parshendi arrows. It's also inferred that the more Words spoken the greater the presence of the spren in the Physical Realm, and the more capacity to Invest Stormlight and retain it for longer periods of time.