Pretty sure I remember City playing Chelsea at home attendance 40k in the top of the table first division clash and United playing Forest in a cup quarter final in front of circa 56k in 1989. Like you say cuts and the world we live in now makes it unlikely both could be played simultaneously.
Around then it was league cup City v Norwich, United v Spurs on same night, when United Played Forrest in the cup at home around 87 we had a Simod Cup game at home against Chelsea I think.
It has happened around 2012 and another time recently too.