As mentioned above you can probably make an improvement through some adjustments.
Basically, you need to shift sooner and/or harder. Lengthen the control pressure cable for the first, run the modulator in a turn for the second.
And yes, the problem is caused by wear. The actual condition is the result of either taking too long to fill the clutch pack (due to more space as the plates wear) or clutch piston seal leakage. Both delay the onset of the next gear.
Bosch Master, ASE Master, L1
33 years MB technician