Contrary to my speculation that Nightcrawler would be CGI this time around, Alan Cumming talked on Today with Jenna and Friends about the make-up process and how much it has changed since last time around (quote courtesy of io9):
Isn’t that nuts? It’s 23 years since I was a superhero…I’ve had some make-up tests already. Actually, what’s great about it was that it was about four-and-a-half to five hours before [to apply], but now it’s only 90 minutes. Before, all of the tattoos were done by hand because they hadn’t decided on them before we started filming, and now they’re on this little thing–they stick on. It’s a game-changer. So, I’m going back to being a 60-year-old superhero.The same io9 article features a quote from Channing Tatum that sort of suggests that he might not be paying Gambit (at least that's what io9 wants to think) but it's all very vague and probably just means he's playing a variant.
One of the reasons Cumming didn't come back for X3 was because of the makeup process being too grueling for him back then. Either he's mellowed out, Disney is paying better or he's fine with a shorter makeup process now.