Below is a quote straight from the Icelandair website under International flights. Note the carry-on weight limit. If your bag is bigger than their limit it may also be heavier than you think it is. Here's the link for all the baggage allowance permutations by ticket class: https://www.icelandair.com/support/baggage/allowance/
Your carry-on bag must fit in the overhead compartment.
Maximum dimensions (including handles and wheels): 21.6 x 15.7 x 7.8 inches (55 x 40 x 20 cm).
Maximum weight: 22 lbs (10 kg).
You may bring on board 1 small personal item such as a handbag, backpack, or laptop bag that must fit under the seat in front of you.
Maximum dimensions: 15.7 x 11.8 x 5.9 inches (40 x 30 x 15 cm).
Here's a recent Travel & Leisure article on flying Icelandair: https://www.travelandleisure.com/airlines-airports/icelandair-reviews
This quote from Tripadvisor reviewer Alicia M. caught my eye: "Note to all travelers — read the emails before the flights, measure your luggage, and weigh it. This didn't seem to be as big of an issue in other countries, but many people were getting bags pulled and charged to check them in the U.S. This is incredibly expensive, so pay attention to the requirements."
As others have said, I wouldn't chance it. I also wouldn't check 2 big(ger?) bags that you'll have to manage on your own in London and wherever else you're going.
You'll find alternate ways to pack for trips of any length and any combination of locations on this website here under Travel Tips: Packing Light: https://www.ricksteves.com/travel-tips/packing-light
And you'll find lots of discussions about those alternatives on the Packing Forum here: https://community.ricksteves.com/travel-forum/packing