I know there isn't a lot of proofs but, I just rewatched Deadpool and I found many clues about when the film is set in the X-Men movie timeline.
By considering that the Wade Wilson from Origins is the same from Deadpool, then we have to assume that the Wade from the latter film is the oldest possible and the one in Origins the youngest possible. If the Wade is, lets say, 25 years old (I think 20 years is just too much) in 1973, when he is in Nigeria, then he would have been born in 1948. If in DP, Wade is 50, then it would be set in 1998. The fact that the events of Deadpool show a world with advanced technology, like cars, gadgets, emojis, etc... the closest the events are from the present the best.
Fortunetly, we can assume when the events are set. During one sex scene, we can hear Vanessa say that it is The Year of the Dog. Well, the years of the Dog after 1973 started in the following: 1982, 1994, 2006, 2018, 2030, and goes on. Well we can eliminate the 1982, because it is set before X-Men: Apocalypse, when the technology doesn't seems to have suffered the leap into modernity. 1994 and 2006 seem likely. 2018 seems to fit, given that the film really looks like it is set in the present, but then we would have a Wade with 70 years old (at minimum), which is far from likely. Not going to mention the 2030 (also because it is mentioned that the Professor X is alive during the events of Deadpool, when he actually died a year before). Analysing the years 1994 and 2006, we can still see that the technology still made a major leap since 1983. But, basing in the age of Wade, we know that he has to have a maximum of 50 (55 if we push to the very limit). If it is set in 2006, Wade would be born between 1951 and 1956. Well he doesn't seems to have 17 in 1973 (or 22 in a matter of fact). Pushing to the limit we could argue that the sex scene is set in 2006, but it doesn't looks realistic. Therefore I bet that it would be set in 1994 and it makes sense, assuming he is only 50 when he becomes Deadpool, two years later, making him 48 when he meets Vanessa (1994) and making him 27 in 1973, which fits very well.
To proof this even more, lets take a look to the references of Wade himself. He references Sean O'Connor, from 1990, in Deadpool No Good Deed he references that the yogurt is from 94 and, during a moment where he is with Ajax, he said he would do to him what someone (I can't remeber who was) did to the music in the late 90's (obviosuly referring to the present and near future for the time). All these references, suggest that the years that Deadpool is set can't be earlier than the suggested date above. But you may notice that there are movies, like the Taken franchise, who were released after the 90's. But, as this is a new timeline, when the events were changed since 1973, the Taken films could be made earlier.
However, there is more things that have to be discussed around here. We can argue that Deadpool is set in 1994-96, but we have to consider the material from the 25 Moments. It is suggested that Colossus was born in 1986 but this guy doesn't seems to have 10 years old. Placing the events of Deadpool in 2006 would, theoretically solve the problem, but we would put him with 20 years old (which would most likely make him more like a apprendice and not a teacher). One of the theories to explain why Colossus appears in adulthood during the events of Deadpool could be explain by the fact that there are two Colossus. One is that who appears in X1, X2, X3 and DoFP, who seems to be American, doesn't have the rings in his arms, and can change from human to metal. The other is larger, older, Russian, has rings in his arms, and doesn't seems capable of transforming himself into human. This first theory seems valid in many points, meaning that i would be the one from X3 that was born in 1986, but doesn't necessarily means it is true (personally I don't think they are different characters). A second theory is simple: the 25 Moments can be considered non-canon. This is because I can only consider the information given by the theatrical release films themselves. But, for those who do, I have a third theory: none of the Colossus (only if they are different characters) were born in 1986. This is because in the 25 Moments, I saw nothing saying explicitly that he aws born there. No name was given to that little metal baby, so we cannot assume that he (she?) is Colossus himself.
Just a crazy theory! Like to see your thoughts!