In reality land, not much. There are probably hundreds of American tourists who were simply unaware or intentionally tried to be smartasses to whom nothing happened. The queen's guards have some authority to detain someone at least until actual police arrives, but at the topmost of the worst you'd be looking at a court date, but no prosecutor would in their right mind bring a case like this to trial, it would be a diplomatic incident if the accused if foreign, or a national debate if they are from the UK.

I guess the worse outcome that could realistically happen is a fine, and even that's stretching it.