Okay my 2 cents worth (that's aussie dollar which makes it about 1.2 cents USA or .7p in the UK).
In Oz, we hardly ever say expedite to an acft to make it climb/descend quickly. Usually the phrases used (and encouraged by ATC instructors and check controllers) are;
- Request best rate of climb, due traffic etc
- climb/descend to ....., reach F... by..... due traffic
The latter phrase is regarded as better.
"expedite" is usually used by tower controllers to make acft move quickly on the ground, eg. cross rwy 01, expedite.
This is because expedite just means do it quickly, hasten. As an instructor or check controller, I believe that most pilots would like to be given a more specific instruction and a reason. Hence a requirement and reason should be given.
Of course, when the poo hits the fan, then the phrase'
"For separation, climb now to F....., traffic is ......." might be used.
DP