A warm up needs to accomplish 4 things.
1. It needs to get the lips vibrating to produce both a
good tone and flexibility.
2. It needs to get the breathing process / breath support working properly.
3. It needs to get the tongue working on single and multiple tonguing applications.
4. It needs to get the fingers to work with the lips, breath, and tongue.
Jacoby used to start a lesson by having the student bend as far forward as possible and breathe with his mouth open (yawning). This helped to relax and open the throat.
Next a secong line g was played until the tone came clear and free. On a good day15 seconds on a bad day as long as it took. Then a scale from the center out by half steps. G, G#, F#, A, F, A#, E, B, Eb, C. Then C,G,E, low C arpeggio. The breath control and slurs were tackeled at the same time. Lip slurs one octave ( no extra notes ) 123 low C#-F#-C#, 13 D-G-D, 23 Eb-Ab-Eb, 12 E-A-E, 1 F-Bb-F, 2 F#-B-F#, open second line G-C-G on top of the staff. This was repeated if extra notes were present or sound was not strong. Next single tongue a F major scale 4 sixteenth notes on each pitch . Up and down one octave until tongue is moving. Last work on the fingers and play a one octave chromatic C scale over and over until it is good.
This is very basic (easy ) music for a reason. It allows you to concentrate on warming up 1 aspect of playing at a time with nothing else to think about. On a good day 5 minutes and you are done. On a bad day 15 - 20 minutes.
A large number of people require more warm up time because their warm up is too complex. They try to think of all areas of playing. If you start with scales you are trying to warm up the lips, tongue, fingers and breathing apparatus all at the same time. This tends to lengthen the time needed to warm up. The warm up I posted will address 1 item at a time.