I think we all agree that Udyr is one of the most classic champions of this game. To adapt his skills so that he can deal with the new champions, in terms of mobility, I've thought about adapting their four spirits to the skills of the four most similar champions (Warwick, Rammus, Volibear, Anivia).
Tiger = Warwick Q
Although Rengar is a feline, the essence of this ability is his claws, which are best represented by Warwick. If Udyr had a skill similar to that of Warwick Q, he could jump on enemies backs and receive some sustain (Udyr W) instead of speed increase.
Turtle= Rammus W / E
The most similar skill to Udyr W is Rammus W, changing the heal by the return of received damage. However, the taut of Rammus E can be more interesting, although given the rest of my proposals maybe it's too much cc.
Bear = Volibear Q
Changing the stun by throwing the enemy backwards wouldn't be a great difference. Although it can be more useful in terms of teamfights and would add some synergy (with Yasuo ultimate, for example).
Phoenix = Anivia R
In this case it would be a slight inspiration, since my goal is only to introduce a little slow in this Udyr skill. Slow that could be better if the damage is AoE instead of cone-shape.
In summary, with 1 or 2 more cc skills and better positioning of this champion by a jump, I think we could adapt Udyr to the current game.