Okay so training wheel champ, little you can do about it, but there are a few things.
So when Jayce runs out of mana, he is significantly weaker, his ranged poke combo is stronger than most ultimate's, Riot has a thing for outrageously stupid ideas like this but just go with it. There is a downside to this combo, it costs a lot of mana, if he OOMs, he will attempt to restore mana in cannon form. As Nidalee, I would take this opportunity to zone him, harass and if he comes close enough all in him, he is not a strong auto attacker, which is good considering he is a strong everything else.
The other thing I can tell you isn't as much help but still, champions to play into Jayce. There are two things you want, and Almost no overlap.
You want champions that can close a gap twice, champions like Irelia, Nidalee, Akali, Diana ect.
However here is where the problem lies, you need champions that on top of the above can win 1v1 against Jayce in an all in. Aka other training wheel champs. Things like Volibear, Darius, and Nasus can be good but lack the gap closers.
Because of this lack of overlap, most Jayce match ups come down to nearly no counter play and it's on the skill of the Jayce, a relatively easy champion, who starts lane with a free win, and it's just about execution, and not throwing it. #trainingwheelchamps
I would recommend playing Zed, Wukong or Akali into Jayce, I have little experience doing it, however I do understand the champions enough that I think these are some of the better champions to go against him.
Hard to knock away Zed if he's untargetable.
Wukong can go in a bush, stealth and then surprise, getting knocked back to immediately use E to close the gap, Idk if he can win the 1v1, but I do know his damage is bullshit enough that mid Wukong is a thing simply due to his ability to half health in a second at level 2.
Lastly Akali after 6 should be able to beat him, she has the multiple gap closers necessary, Idk about the power, issue is surviving until then, but that's most Akali match ups.