Cineuropa has a post up on Nicolas Winding Refn and the fact that his The Neon Demon is the only Scandinavian film that will be competing at Cannes this year. the Danish release of the film will be on June 9. A little latter in the article they say the following:
Supported by the Danish Film Institute, The Neon Demon will be launched simultaneously in the UK, France, Italy and the US (where it will be handled by Amazon). Refn, who is celebrating his 20th anniversary as a director, has thus delivered the Danish feature set to receive the largest international opening of all time. (bolding mine)
That seems to indicate that the film will also be released in the listed countries on June 9.
Vogue Italia has a post up on the film and at the very end, it says the film will be released in Italy on June 8.
IMDb is showing a release date of June 8 for France, a release date of July 8 in the UK and a release date of June 2016 for the USA. Based on the info in the article, you have to wonder if the date has been set for the USA release and the date moved up for the UK release.