This is a kind of fun idea, but a lot of the combinations would make for pretty boring television.
The particular relationships between the individual characters in each show were the dynamic that made each one unique and entertaining in its own way. For example the antagonism between McCoy and Spock being balanced out by their mutual respect for and friendship with Jim. So even though I love, for example, Data, his presence in this dynamic would ruin it.
Having said that, here's one that I think would make a terrible show:
First Officer: Wesley Crusher
Second Officer: That annoying barber guy with the blue head on the next generation
Tactical: Will Riker
Navigation: Thomas Riker
Security Officer: Paul Blart: Mall Cop
Science Officer: Nurse Chapel
Ship Counsellor: The Crystalline Entity
Red Shirt: The Outrageous Okona