If you read ancient books or historical textbook, 京城 may refer to the ancient capital(such as Chang'an) depending on time. But now we only use it for Beijing.

"京城" (or "京师") is an archaic form of "首都", just as "thy" and "your". In this sense, you are right about "monarchy". "京城" was not only used in China but also in those countries influenced by Chinese culture in ancient times. Thus, in ancient Korea, it referred to Seoul and in ancient Japan referred to Kyoto. Nothing prevent you to call London as the "京城" of UK. Just a little bit old-fashioned. Generally you say "英国的首都伦敦" while "大英帝国／日不落帝国的京城伦敦".

京城 = literally imperial city/castle 首都 = capital, literally "chief city" 京城 can be used for Tokyo (東京) because Japan still has an Emperor. 北京, 南京, 京都 (Kyoto) are still used because of historical reasons but are not the "chief city" So unless the country is an Empire styled government, 首都 is used. England is a kingdom but never an empire in the true sense of the word but a kingdom of colonies, protectorates etc; So London is just the 首都. Moscow can be used as 京城 in Tsarist times. Here's an interesting historical fact. Queen Victoria cannot have the title of Empress because the British loathe the idea as an empire has a connotation of autocratic rule. Her daughter (also named Victoria) married the German Emperor and became Empress. Queen Victoria did not like the idea of having the daughter to be higher ranked than her. So the English Parliament's way out was to give her the title of Empress of India. King George, her son also inherited the title of Emperor of India when he became the monarch.

京城=北京 北京是首都 伦敦London是首都但不是京城 London is 首都but not 京城

京城 is an old-style way to call ancient Chinese capitals, just like "thee" and "you" in English. Modern Chinese people don't say"京城” to indicate capital. It's a "very Chinese" word belonging to the past. So never say 英国京城，it doesn't make sense.