Re: cough up
In my experience, when you sell someone down the river, you betray them in a non-relationship sense.
There has been some sort of wrong doing. One person gets out of trouble (or in less trouble) by betraying the other person involved.
Two crooks are arrested, and one tells the whole story in exchange for a lighter sentence, in effect selling his buddy down the river.
(I even wonder if it has to do with the location of London Tower -- did you sell someone down the river so that they ended up in the Tower? At first I thought of SingSing, but that's upriver from New York City. My London geography is terrible, so I don't know if that works for the Tower.)
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.