The Blues have been chasing Cuadrado all month and while it looked likely to happen, there were a lot of contingencies. Most notably, Chelsea had to sell Andre Schurrle to Wolfsburg and then make the Mohamed Salah loan to Fiorentina work before they could buy Cuadrado. With the Schurrle deal going down to the wire, there was a possibility that Cuadrado wouldn't get done, but the Blues worked out the Schurrle sale, clearing the way for the Colombian to join Chelsea.
Salah still hasn't been completed, but Chelsea and Fiorentina got the deal done without the Egyptian's loan.
It would have been awkward if Cuadrado did not switch to Chelsea. He had posted a goodbye message to the Fiorentina supporters and talked openly about playing for Chelsea. Had Schurrle or Salah prevented him from jumping to Stamford Bridge, going back to Fiorentina may have caused some strange scenes.
Schurrle struggled this season while Salah never really cracked the squad at Chelsea, so the Blues needed to make an upgrade for the second half of the season as they chase the Premier League title. Cuadrado does that for them, giving them added depth and an excellent player.
Chelsea already had a starting front three of Eden Hazard, Oscar and Willian. Now they add Cuadrado to the mix. Once again, the rich are getting richer.