Some other questions to consider are how much of the British Isles are you wanting to cover on your British Isles overview -- western, southern, and/or Northern Ireland, in addition to (or excluding) Dublin; Scottish Isles, Highland moors, cities, Southern Borders; Wales; England? We just got back from 17 days in just Scotland (and needed more time, especially with the August rain that limited some of our activities), and flew from the USA to London, then directly to Edinburgh (1-hour flight). A train would've gotten us there slower but woul've afforded a chance to see parts of England along the way. After driving and ferrying around Scotland, we flew Edinburgh-London-Denver, and were up for 23-hours, straight, yesterday. Getting to London by land would've added much more travel time and eaten into overall sights and activities.
If you were trying to minimize any backtracking, and flying back over a place you've just been isn't the same thing as zig-zagging all over the place on land, so doing a stricly linear journey may not be as critical for you if you're flying around than if you were doing a bicycle or walking trip, but if you flew into, say, Heathrow, you could hop onto the next flight bound for Dublin or Shannon in Ireland, then work you way over to Scotland by plane or ferry, then down thru Wales/England, ending up in London proper. Your sightseeing priorities could determine how much time you want to spend in a given location, and are you planning to travel by car or public transportation once you land?
You can even take a ferry from Harwich, northeast of London, to the Hook of Holland, so that could be an option for getting to Amsterdam. In addition to the cheap, budget, minimal-frills airlines mentioned by David, others such as easyJet and Vueling also can get you around Europe quickly by air (although you have the security, baggage, and check-in time requirements that train, bus, and car travel don't), so those could give you some possibilities. Night trains can be efficient, but don't guarantee a great night's sleep. Day trains let you see the countryside passing by, but don't put you on the ground, doing things at your choosing.
Back to Scotland -- we didn't opt for any guided daytours, but lots of minivan and bus tours are available, and might work for seeing sights without the challenge of driving narrow roads on the left, with other tourists hurtling at you in the other direction, crossing over the centerline into your lane! :)
For what it's worth, after several days in Edinburgh, we flew (flybe Air) to Shetland, took the ferry to Kirkwall in Orkney and got another rental car, flew to Inverness, then drove to Skye (watch out for the French, Belgian, Italian, and German motorhomes and cars also on the road, but not necessarily sharing it), and ferried/drove back to Edinburgh. We hope to return -- hope your trip is great!
