I don't know an exact year or range of years but I think they were distributed in the UK exclusively by Peradon. You should start by reaching out to them, and I will continue to have a look on my end.
I am going to ball-park it somewhere in the early 1980's (but possibly late 1970s) and here's why. Cliff Thorburn had professional status as a snooker player as of 1976, and was highest ranked snooker player from 1981-1982 (#1). It is likely the cues were done some time around or after this, as Thorburn was a top-ranked professional snooker player all throughout the 1980s.
Here is a Riley Joe Davis cue (manufactured "by Dufferin") and that "by Dufferin" stamp is also stylized the same way. It was listed as a 1980's Dufferin Snooker cue, and this kind of confirms my thoughts above. Not sure how accurate that is though.
1980's 2-Piece Riley Steve Davis Snooker Cue "by Dufferin"
Also, I have seen similar (but not identical) Dufferin Joe Davis Signature cues, which has a "Dufferin Cue" logo stylized the same as the "Dufferin Pro" in your cue. Thus, we could say they are likely from the same era. For this one, the guy who posted it said it belonged to his Grandfather. That is a vague indicator of the cue's age.
Give it a few more days, there are some other guys on here who may be able to provide additional detail on your snooker cue.