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The 'off licence' was a part of the pub that was able to sell alcohol without going into the bar. Some may remember them as 'outdoors' or 'offys'. Age did not apply, as it did in the bar. So children would take a jug to the serving hatch in the pub and get beer for dad .
The licensing act 1964 created the off licences that we know today, either in the High St or part of the supermarket. It applied age rules which stopped children buying alcohol.
Looking at the display window in the photograph, it looks as if it may be a wine and spirit dealer. I have seen an advert for Foulkes and co in the Old Town stores. Try this link.

  .shttps://rhylhistoryclub.wordpress.com/2015/04/19/victorian-and-edwardian-bottles/

Yes I thought it might be the Bee. I went there this afternoon and it is a different design. There are pictures of the Bee Hotel on the internet. The Lorne on the junction of High St and market St. has a similar pattern which has now been painted out. Could not find anything in Queen St nor Water St.
(Most of the buildings in Rhyl town centre are still original.)
141 Wellington Road is about a mile away from the town centre, in the direction of Kinmel Bay (roughly where Aldi is today.)
Did Off licences in those days sell beer 'on draught' ?
I am told that there is site of Facebook called 'we love old Rhyl' Perhaps someone on that site could help.
The White Horse pub in Bedford Street, looks as if it was glazed tiled. No sign of that today, the outside has been modernised.
Not much help I'm afraid. The term 'off licence' did not come about until the 1960s, so I am inclined to think this may be a pub, or part of the theatres that existed on the promenade, which have all been pulled down.

Eldon,what do the numbers alongside the Conductors relate to?I'm sure my brother's name features.
Regards#
William Russell Jones.

William, It is the number of appearances made in competitions. I have sent you a PM.
Regards Eldon.

Have a look at Bagillt High Street on Google Maps.

https://www.google.co.uk/maps/@53.262296,-3.159869,3a,75y,180h,90t/data=!3m4!1e1!3m2!1sjvpVeQ_-eR4Afc5mfFRKtg!2e0

If this shows a picture of a white farm house then you are one end of the village before it re-joins the bypass. Rotate the camera (bottom right) and travel through the village. A great deal of the village has been knocked down and rebuilt but you will see 'The Blossoms' and past the community centre you will see 'The Stag Inn', opposite the 'Bagillt Arms Hotel' (boarded up) Further along you will see 'The Feathers' and that's it. The road joins the bypass.
The Jehovah 's Witness Church is set back amongst some blocks of flats, opposite and just a bit towards the village centre. Their car park may be the location of the Wellington.
Hope this helps.