What I did in London Transit last year during layover was headed toward the departure, lined up in immigration, told the immigration officer that I have GC and 11 hrs layover and wanted to roam around London, After looking at my GC, she stamped me a 24 hr temporary stay status (not visa ) in my passport and I had my fun in the city. 

I had made up my mind to stay in the transit even if they had turned down my request which they did not. In short, there is no visa application in London transit. All you do is go to the immigration in normal departure and request to allow u to enter the city.

You can apply For UK visitor visa in US in case u don't want to take risk of getting rejected to go out of terminal but it takes time for prepartion, application and money. 24 hr stamping in London immigration was free.