I've just been going through the old press cuttings for clues, and the best I can offer is an undated Kerrang! news piece announcing that The Wildhearts would be supporting Izzy Stradlin And The Ju Ju Hounds on their UK tour "beginning next week". At the end, it mentions that The Wildhearts "have just released" Don't Be Happy... Just Worry.
The
Ju Ju Hounds tour was due to reach the UK on 4 December 1992 (the first three dates were apparently cancelled, but as that probably happened after Kerrang!'s print deadline I don't think it matters). So if the tour was "next week", and the EP had only just come out, it looks like its release date was closer to the end of the month than the beginning - though this theory depends how literally you take the term "just released". It might narrow the field, but it might be a red herring; I don't know.
The EP got 4Ks in Kerrang!, by the way. Steve Beebee did the honours. As Damian says, if anyone does have the original magazine (I don't - just the page), the date on the front will be a strong piece of evidence for the record's release the following Monday. I think that, generally speaking, Record Collector's dating style just lists the month and year, though I might be wrong.