Travel Itinerary

Start by reading Europe Through the Back Door by Rick Steves to get yourself oriented around this entire topic. On your schedule you might visit the capital of each country and make one or two day trips to sites within 3 hours of travel.

Three months seems like a long time, but you also have an extensive list of places to visit! This seems like a daunting task to plan such a great trip! One thing that might help is to break this down into a few regional trips of about one month each for planning purposes. You can group as:

1 England/Wales, Scotland, Ireland, France
2 Norway, Sweden, Finland,
3 Denmark, Germany, Poland Austria, Slovakia, Czech Republic, Hungary, Romania (maybe)

For the first group, you can allocate your month by starting in Ireland, then flying to Scotland, then hitting Wales/England. I included France in this group because it's an easy train ride from London to Paris. If you split your time evenly, you would have about a week in each of the four locations. You will want to adjust your time in each location based on your interests and preferences. You may also find that you want to drop some places to maximize your time in the places you are most interested in. From a logistics/travel persepctive, it may be best to go from France to Germany and the rest of group 3 and then finish up in group 2.

If you look at the RS tour dates, you can probably do a few back to back tours. For example, you could link the Best of England with the Best of London and the Best of Scotland. If you are interested in this idea, I suggest contacting RS Tours directly and try to work this them on an itinerary. As you suggest, you will only have to buy one airfare. As an example, if one tour ends on Saturday and the next one starts on Tuesday, you will have a couple of extra days to explore where you are or travel to the start of the next tour.

Good luck!

You probably know this, but just in case.
You said "approximately 3 months". But, as a non-European tourist, you are only allowed in the Schengen Area countries for a maximum of 90 days in 180. That is 90 days for all the countries. The UK is not in Schengen, neither is Romania (but it may be by next summer). but all the rest are.
So long as you spend less than 90 days in all the other countries you list, and anything over in the UK (which counts separately), you will be OK.

Otherwise, have fun. Get a map of Europe, put a mark on all the places you want to visit, join them up in the shortest way, write a number of nights against each, and start planning.

That’s a lot of territory. You might rethink about maybe setting up camp in half a dozen places and then doing shorter trips from those home bases. A new hotel every few nights would wear me out. In between the bases you might do a few one night stays to shorten the road time.

An idea for instance might be to get an apartment in the UK for a couple of weeks and then from that apartment do some short trips, overnight even, around England, Scotland, Wales and Ireland (actually Ireland is probable a good stop on the way over).

You didn’t mention means of travel preferences or budgets so it’s hard to come up with ideas for you.

I'd start with making a list of more specific places you're interested in. France - great, do you really want to see Paris? Normandy? Provence? Germany - Berlin? Munich? Bavaria? You get my drift. You could hit a place in all the countries listed in 3 months, but if you have a very long list in one country, that will take more time. List your places, then make a map (either draw on on a paper map, or do a google map) showing where all the places are. From there you can narrow it down.

Rick Steves' Tours don't include airfare, so you always do book your own flights, making it simple to combine tours or add personal travel afterward. Descriptions online for each tour include details for Itinerary and Activity Level.

For independent travel, you might consider a long-term car rental, also called lease, under the Peugeot or Renault programs administered from France. These allow pick up or drop off in neighboring countries for a small fee. I did pick up and drive a French car in Britain with no problem (except the problem of toll booths being on the wrong side of the car). Driving across this many countries would be a lot of kilometers, so that plan works better if you'll make more stops in fewer countries.

Or you could rent separate cars for various regions. Note that regular rentals can have high fees for dropping the car in a different country. If you're thinking of rail travel, you'd consider more than one rail pass - probably one BritRail and one for the continent, such as the Eurail Global. That's a decision that can come much further along in the planning steps. Some longer legs are pretty affordable to connect by flying, such as from Britain or Ireland to Scandinavia. See also http://www.ricksteves.com/travel-tips/transportation.

I bring the same amount of clothes for two weeks or several months, with the plan that I will visit a Laundromat every 2 - 3 weeks. A rental car can carry some items that you don't always need to bring into hotels, such as extra guidebooks, but when you travel by train you will typically carry your own gear. If you need it, elderly or disabled passengers can often reserve ahead for assistance to board trains.