While Warwick seems like a champion that would scale very well with upgraded boots, this is untrue. In most cases Warwick does not need to upgrade his boots as there are items that synergise better with his kit, which are a priority. However, there are 2 cases when boots might be very effective to get by mid-game:
1. If the enemy team has 5 AD champions, or 4 ad champions and at least 2 are autoattack reliant, get ninja tabis
2. If the enemy team has 3 AP champions or a lot of crowd control, get mercury's treads
If it gets to very late game, don't forget to upgrade your boots after you have finished all of your other items, generally go for mercury's treads for team fights.

I have played Warwick constantly ever since his rework came out and I have been top 300 Warwick players in the world (according to lolskill.net) in less than 100 games, plus I had a 65% win rate in diamond for until my account got banned. If there is someone that you can trust with a Warwick guide, it's me.
For reference, TheDoggyHacker EUW.