It's so easy to travel to London that it wouldn't be my top priority here.
I am not a meteorologist, but I've done some research on weather before each of my recent trips to Europe in an effort to pick the right destinations for the time of year I'm dealing with. In general, the western edge of Europe (Bergen in Norway, western Scotland, etc.) is wetter than places farther east. So Oslo is likely to have less rain than Bergen, and Edinburgh less rain than Glasgow. Unfortunately, some of the most dramatic scenery in Norway and Scotland is in the west. Few people would read a guide to Scotland and choose to see only Edinburgh, or a guide to Norway and choose to see only Oslo. It's clear from your post that you'll want to spend a great deal of time outdoors.
The Wikipedia articles for most cities include a climate-summary chart. You may have to click on "Show" in the "Climate" section to see it. That chart usually includes some useful statistics related to precipitation. I like to compare that data for cities I'm considering to each other and also to the same information for my hometown. Perhaps that will help you make a decision.
For example, Bergen gets an average of over 10 inches of rain per month in September and October, with rain falling on an average of 17 days in September and 19 days in October.
There's a lot less total rain in Glasgow (3.9 inches and 5.2 inches), but there are still an average of 13.9 rainy days in September and 16.2 in October.
That far north, days will be getting noticeably shorter by the time you leave Iceland, no matter how the weather turns out.
There's a real possibility of weather chilly enough that it wouldn't be fun to get soaked to the skin. You'll see that the average low temperatures for September and October are down in the 40s(F). My trips to Norway and Scotland were in June and July. I dealt with my hatred of chilly, wet weather by packing merino wool long johns and nylon slacks that shed water fairly well--in addition to a waterproof jacket. The trips were long and not expected to be wet throughout, so I didn't want to go so far as to take rain pants. That's something I would do if traveling only to Iceland+Scotland or Iceland+Norway in the early fall. But I am a weather wimp.
I have done no research on Ireland but suspect the same caveats apply there as to Scotland.