Ireland, Scotland and northern England

Yes. By a long, long shot. You had your suspicions before you asked the question, right?

You could do it. You would not get to explore any of those locations 'in depth', but could get a taste of each. It might would something like this- Fly into Dublin- spend 1 or 2 nights Rent car and explore Ring of Kerry- 3 nights Fly from Dublin to Glasgow (if possible- I'm not checking flights)spend 2 nights Train to Inverness- 3 nights Train to Edinburgh- 2 or 3 nights Train to .... Not sure where in northern England you want to visit
As you can see, it is rather a fast pace and you are only hitting the highlights of a few places. 2 1/2 weeks would give you very nice amount of time in EITHER Ireland or Scotland. Or Scotland and a bit of England.

Yes, 2.5 weeks is enough to get a nice taste of each country, as long as you limit the number of cities/areas you want to see in each. Toni's suggested itinerary makes a lot of sense.

My only suggestion from above advice is to fly into Shannon if possible, explore Dingle, we prefer that to Kerry but that is only our opinion and then go to Dublin and on to Scotland, do not think it is a wise to fly into Dublin, go west to either Kerry or Dingle and then drive back to Dublin. It really depends on what flights you can get to Ireland and home for Scotland. You are looking into open jaw aren't you?

To add to the conversation, here's some points to consider: 1) In Inverness, you can rent a car and see Eileen Donan Castle and Isle of Skye. You can skip loch ness (pleasant, but not a must see). 2)Alternatively, you can skip Inverness. Stay in Edinburgh and day trip to Stirling. You can also take a one-day tour with a company like Rabbie's in Scotland to fit in some nice sights. 3)Then, go to North England. See Alnwick Castle and gardens, York, Hadrian's Wall and whatever else you can. I can't tell you much about Ireland or England. Perhaps, start in Ireland. Get to Edinburgh as a Scotland base, then North England. 3)I think it's doable. You will just have to be somewhat ive, set priorites, and plan your time well. Figure out flights, train transportation, car rental, etc. That will make a difference. You can even "steal" an itinerary from a tour company by researching online. (Try Rabbie's/Rick Steves or other travel company itinerary.)
Most of All - Have fun and enjoy. If it's too stressful cut back on locations.

What time of year is best for this?

Best time of year? Go to RS Tours elsewhere on this web site. Pick out either Ireland, Scotland or England tours. In the lower right hand corner of each you will find something that says something like "Tour Weather".
Click on that for each one to get a yearly report on that places average weather. I have been to all three places and was glad I did each in the summer as it was only mild then but not as cold and wet as other times of the year.

We did a week in Ireland and a week in Scotland a few years ago, and with a few more days, I think you could see some of northern England. I am planning a trip there myself, and from what I'm reading, you could get a taste of Yorkshire and the Lake District in about 3 days. Just have to be willing to do some one night stays. At the time my kids were 7 and 10 and they really enjoyed the trip, but we had to take a slower pace and limit one night stays. It was still a whirlwind tour!

In Ireland, we spent 2-3 days in Dublin, drove South to Wicklow and then out west to the Dingle peninsula. Stayed near Tralee at the Ballyseede Castle--loved it. Stayed in Ennis as a central spot for surrounding areas (Cliffs of Moher). Then flew to Edinburgh, probably my favorite city ever! One of the Scotland highlights was staying in St. Andrews for 2 nights during the week of the British Open. We were there for the practice rounds and my son got ten autographs including Tiger Woods! He and my husband are golf fanatics, so if you have any of those, don't miss the town. Or even if you don't because it has some fantastic historical sites as well. We drove up to the highlands, did some island touring based out of Oban, and came back through the center, flying back out of Edinburgh. If I could go back, I'd probably fly into Shannon, tour the west first, then go east to spend a few days in Dublin. We probably wanted to avoid drop off fees for our car. And if I had a few more days, would have definitely gone to Rosslyn Chapel and down into England.

Good luck and have a wonderful time!