My traveling buddies and I had to postpone our month long trip to England until next spring. We will be visiting Edinburgh and Wales also during that time. What I’d like to consider is a quick three day trip to either Ireland, Denmark, the Netherlands, or any of the surrounding islands. This is not a first trip so we will have time to fit this in and have already decided to allot a few days for this trip. I also know that we can’t fully explore another country in that time. I’m asking for suggestions about which to visit for a short break. I have been to the Netherlands so I’m going to compare my experience to those in your suggestions. Thanks for your suggestions!
Not being a fan of the sort of weather I'd expect to have in England, etc., at that time of year, I'd head as far south as possible. Sticking to the areas you've specified, I guess that would be Jersey or Guernsey in the Channel Islands. One can fly from London; I haven't checked on other flight possibilities. Either of those islands would offer enough to do to fill more than 3 days as long as the weather cooperates. I last visited (Jersey only on that occasion) during the summer of 2018, and I did spend a lot of my time outdoors.
I liked Glasgow just as much as Edinburgh, though the two cities are quite different. Glasgow has a lot of indoor attractions. However, its location toward the west side of Scotland makes it (usually) wetter than Edinburgh. Glasgow is particularly recommended for folks with an interest in architecture. I had a great time tracking down the Charles Rennie Mackintosh buildings.
Not sure why you want to waste all the travel for a short trip?
I would pass on the continent and if you insist go to Ireland. Depending on where you leave England/Scotland/Wales, you could visit Dublin or Belfast.
We did a wonderful 4 week drive tour of S. Wales and England in 2017.
Here is my review:
28 days in Britain and Celebrity Eclipse home https://www.cruisecritic.com/memberreviews/memberreview.cfm?EntryID=599139
A month is nice. :) Still not a huge amount of time but it depends on how many different areas of Scotland and Wales you plan to cover. My answer would be to do some research and just see what appeals to you the most. You could see a good amount of Dublin with 3 full days (as opposed to 3 nights, which would mean only 1 full day and possibly 2 partial days).
Going to think out of the box and say fly London to Milan. Do this at the end of your sojourn.
See the Last Supper in Milan then train to Venice and spend 2 nights.
Then take the high speed train from Venice to Rome and fly home from Rome. Give yourself one night in Rome.
Or fly Manchester to Dublin. A night in Dublin, next day drive to and explore Glendalough and the Wicklow mountains. Another night in Belfast. Early next day train to Belfast and spend the day. next Train day back to Dublin and fly home from Dublin.
I'm going to throw in something completely different. Have you thought of visiting the Isles of Scilly? You can fly from Bristol or Exeter, I believe (not too far from Wales) or get a connecting flight from Cardiff. I've never seen any discussion of Scilly on these forums and it seems like a pretty cool place if you're interested in island landscapes, birds, or just being able to say you've been somewhere rarely seen by Americans!