For Microsoft Copilot, simply visit https://www.bing.com/chat?q=please+introduce+python+to+me+briefly&sendquery=1, where the parameter q is your prompt. (Although it doesn't seem to "start a new chat" as you've requested)

Since I personally do not use Claude, I'm unaware of its shortcuts. However, you can always write a UserScript that checks for your prompt in the url and inserts it into corresponding chat platform.