Back to realityville, what comes to mind is something like this. It requires a much more cynical and bitter world, especially in Europe, after the fall of the Iron Curtain.
1. Thatcher and Reagan are at best rivals, not friends. Thatcher probably can't be butterflied away as PM, but a serious falling out or (never falling in) with Reagan is possible. Say the US provides major opposition to the Falklands Campaign.
2. After the end of the Cold War, Britain goes heavily towards the EU and tries to become a leading power in it. In the process, the end of the Soviet threat leads to a re-evaluation of American relations, and Britons grow distrustful of being an American proxy. Conservative and Labour platforms build towards almost icy relations with the US.
3. Evaluation of American actions during the Cold War also lead to a general anti-Americanism throughout Europe, that at best, America was a lesser evil than the Soviets. Relations turn downright hostile, with a broad withdrawal from NATO. A combined EU military and nuclear weapons program is built to establish that they will not be intimidated. This turning from the US does not drive them instead into the arms of Russia - instead Russia is treated with distaste and even faces significant energy tariffs, as the EU negotiates an excellent deal with OPEC that ensures the flow of oil, while indirectly supporting the foreign policy in number 3.
4. Muslim terrorist attacks are focused on the US and Russia. The strategy is to attack the two "Great Satans" head-on, with the idea that the rest of the West should be left alone to isolate the Eagle and the Bear. Even the most extreme sects are warned not to commit violence or any sort of threatening behavior towards distant countries other than Russia or the United States (agitating against Shiites in Iran is fine, because it's a turf fight, not an anti-Euro fight). For example, around the same time as the 1st World Trade Center bombing, there's a successful bombing of the Hermitage, destroying many priceless Russian cultural treasures. World reaction is basically that the Soviets deserved it for Afghanistan and the US deserved it for its Middle Eastern policies. The US and Russia don't trust each other but share resources in identifying and attacking Al-Queda. Successful operations lead to greater openness and trust in the future.
5. 9/11 attacks happen as in OTL, but there are also bombings or crashes into the rebuilt Hermitage, St. Basil's Cathedral, the Kremlin, and along several major oil pipelines (Wall Street for the US and Oil for the Russians).
6. The UN rejects either US or Russian action against the Taliban thanks to security council vetoes by Britain and France. China abstains, owing to economic pressure from the Eurozone, which is courting China much like the US has been doing. Britain and France, both led by highly anti-American governments say that these asymmetrical actions do not affect the world as a whole, and that they will not send their sons to help Americans and Russians dig out of the hornet's nest they brought upon themselves.
