Let's look at the good deal first and see how you want to deal with that.
. On the front end, Scotland makes the most sense - - either Glasgow or Edinburgh. That'll set you back about a hundred bucks each on Ryan and you could get there in an hour from DUB - - by noon if you go straight with an early morning Dublin arrival. Figure the better part of a day, door-to-door, if you're not already at the airport.
. London's on the far end of the stick and you need to get back to Dublin for a flight, so you'll have to scram the day prior - - another whole day down the drain and another hundred bucks each.
If you're up to it, you can drive out to Killarney the day you land. It takes, minimally, all day to drive the ring. Figure another day to get back to Dublin and catch a late flight to Great Britain.
You can get a good taste of it in four days. If you pick it as a hiking area, I can help since I've walked the whole perimeter and most of the middle.
You're going to have to pick and choose. I've spent close to a year mucking around the area and am still not quite done.
Here's a pretty good list of the games: http://www.shga.co.uk/visitor-events.php. Ten days could give you a pretty good taste if you stayed off the far islands. I've done a lot of hiking there as well - - it can range from the Cape Wrath to easy walks in the Trossachs.
Five days is about right if you stay out of the art museums.
The Rest of England.
Again, pick and choose. A couple of years couldn't polish it off.
Hiking is easy in the Lakes, not too bad in Peak District, lots of variety in Cornwall and Devon. Not to mention a couple dozen areas that are just as good.
I'd think it'd be hard to fit in both the RoI and Wales along with the rest in just three weeks minus the logistics time. If you decided to skip Scotland instead, you could fly from Dublin to Manchester and then work your way south.
It'd help to know how you're planning on getting around. It'd also help to know what you consider hiking - - how many miles a day, how much elevation change is okay, how many days in a row.