I live in Leicestershire and so would not be overly thrilled if the airport changed its name to just Nottingham (although if I was EMA the amount the people of Leics complain I wouldnt be surprised if they didnt want to mention Leics at all!)
However, as someone with many contacts in Spain, I can fully appreciate the fact that very few people actually know what "east Midlands" is, as this is a phrase only used in the United Kingdom, elsewhere this phrase is little known.
But, if you do mention the name "Nottingham" people know what you mean. It is a place many Spanish people visit (the first thing I always say is "why!?") because of Robin Hood, the Forest, the team, the forest, the castle and apparently "its amazing shopping centres"
So ... from a tourist perspective it might be a good idea to include Nottingham in the name.
Maybe a Leeds/Bradford type thing: Nottingham/East Midlands, Nottingham/Derby/Leicester, East Midlands serving Nottingham ...