TL;DR: Late 60's/Early 70's
As you stated, an exact time-frame is never given. However, we can narrow it down a bit. We know that it was after WWII and before 1975 (which is when Wolverine left Department H and joined the X-Men). We also know that Logan escaped Weapon X several years before that event, having wandered the woods for at least several months before the Hudsons found him. They then spent several months nursing him back to health (and sanity) before convincing him to join Department H, and THEN he spent at least a year working for Department H as "The Wolverine".
So, let's say that it was two years from the time he escaped to the time Professor X recruited him, which is probably a conservative estimate. That means he escaped at least by 1973. On the other side of things, we know that Team X - which came first - began in the mid-to-late 1960's, via this entry in the Marvel wiki:
The first Weapon X project took place in the US in the 1960s,
sponsored by the CIA. They recruited skilled intelligence operatives,
who became Team X, made up of Logan (Wolverine), Silver Fox, Victor
Creed (Sabretooth), Christoph Nord (Maverick), John Wraith (Kestrel),
and Mastodon. Although successful at first, the team was disbanded
after a mission involving Omega Red.
Based on the assumption that Team X started in 1961 and then lasted a few years before disbanding, we can safely narrow down the Weapon X project (or least Logan's time with it) to somewhere between 1965-1973. This also corresponds with the clothing & equipment seen in the "Weapon X" storyline by Barry Windsor-Smith, all of which has a late 60's/early 70's feel to it. We also know from various flashbacks that Logan participated in The Vietnam Conflict at some point, so he was free for at least SOME of this time prior to his kidnapping. Factoring that in, we could probably narrow Weapon X down even further to just the time around 1970-1973. Unfortunately, flashbacks by themselves are unreliable in Logan's case, so we'll leave that part to speculation.
ADDENDUM: Note that if reliable, his time in Vietnam could have conceivably taken place as solo missions while he was with Team X. The CIA had operatives in Vietnam as early as 1964, and probably even before that, so the timing does line up fairly well to support that idea.
UPDATE: I did some further digging and found this on the Marvel Wikia. Emphasis is mine:
In the 1970s the Weapon X project expanded into Canada, where it was
run by Department K but still funded by the CIA. The goal this time
was to enhance the abilities of agents already working for Department
H, including Ajax, Deadpool, Garrison Kane, Sluggo, and Slayback.
Their test subjects did not fare as well as Team X, however, and many
of them were permanently disabled. Even worse, these "dropouts" wound
up at the Hospice, where Dr. Killebrew and Ajax performed unspeakable
(and unauthorized) experiments on them. After Deadpool led an escape
from the Hospice and exposed Killebrew, the entire Weapon X project in
Canada was dismantled. Dr. Horatio Huxley revived the project
The American project, meanwhile, was dusted off and revitalized. The
staff included "Professor" Andre Thorton and John Sublime (both
veterans of the Weapon Plus program), as well as Dr. Carol Hines, Dr.
Abraham Cornelius, and Dr. Dale Rice. Thorton located Wolverine and,
in a secret lab (near Roanoke, Alberta province, Canada), augmented
his skeleton with adamantium. Temporarily insane from pain, Wolverine
massacred most of the lab personnel and escaped. Thanks to this
setback, Thorton's Weapon X project lost the support of the Weapon
Plus program, but Thorton continued his work.
So, it seems that "In the 1970s" is probably the best answer we'll get.