Annoying war-score issue

UK will mostly intervene in any war if 1. it is friendly with attacked country and 2. you seem to be an easy target.

About warscore, the AI will sign when they think they can't handle the war anymore. Since UK intervene and he is the warleader, he will not give a damn about Egypt until you screw it instead. By a rule of thumb if UK is involved in a war against you, you better be able to occupy UK mainland or a white peace is almost inevitable...

They intervene because Egypt is close enough to be in their sphere and they'd rather limit your expansion, seems reasonnable.

On the warscore, it's possible to enforce wargoals against the UK without occupying their homelands, it just takes a bit more time and it realizes it won't win. You can't expect a +10% bonus in war score to force a negociation because you could just jump on a few scattered armies and accumulate warscore in a few months, while it's understandable that a Great Power would be more resilient than that.

It's also understandable that Egypt doesn't want to give up, when it's being backed by so many powerful allies.

Put it this way - imagine yourself in the spot of UK, you are the world's #1 power, why would you not meddle in everyone's affairs and dictate how the world is shaped? I, as the player, would definitely be all over that (and often am) - an aggressive player would always want not just conquests for himself, but also to keep his friends strong, his enemies weak and to keep a balance of power in specific regions he/she is interested in. In this case, UK and other countries want to keep the balance of power between Egypt and yourself, while Russia is just evil and hates you by default.

US is probably interested in Suez Canal just like UK and they'd prefer it to be covered from both sides with friendly Egypt (since you need to have them sphered to make it anyway), not a potential rival.

You say it's bullying. Well, it pretty much is, but it sounds reasonable to me, given the context. The eastern Mediterranean was quite an issue in those days. When it seemed like Russia had a chance of becoming a naval power there, Britain and France intervened. Now, if the Ottoman Empire rose to strength again and became a naval power in the eastern Mediterranean, Britain would have definitely smacked them down at some point. And Russia would have joined in because picking on the Ottomans was Russia's national pastime. Also, Egypt is pretty strategic. I can understand why other countries would be keen on not letting you have it.