This is great. Taking nothing away from the question or the excellent answers.
Shottky Baker clamp and Speedup Capacitor were really important in the 1970/80s.
I had a similar challenge recently. Trying to drive NeoPixels (WS2812B) at 7.5V from a 5V MCU, needed a level shift up to 7.5V to make the first NeoPixel in the chain read the logic high correctly. So I needed a fast switch for the level shift.
I was running the NeoPixels at 1Mhz, but the short pulses (which mean logic zero), are only half as long , so the effective frequency is more like 2 or 3Mhz. So even with both, a Shottky Baker clamp and a "Speed up Capacitor" , I was still struggling to level shift the NeoPixel waveform with a BJT without distorting it badly.
However, perhaps the "most standard small signal MOSFET" out there, the 2N7000, costs pence these days, comes in all sorts of packages including a through-hole TO92, and eats this speed for breakfast. I ordered ten for the box, plugged one in and the problem was solved. The turn on/off times of the MOSFET are faster than what the MCU was able to put out, so the waveform was perfectly preserved.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/RALmo.png)
Same Circuit (minus Shottky and any SpeedupCapacitor).
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/mDDQV.png)
You can drive the gate of the 2N7000 directly from the MCU, or use a ~470Ohm resistor to "protect the MCU". Choose a pullup resistor on the Drain of the MOSFET which suits the capacitance of your load.
The moral of the story is: BJTs are not very fast at on/off switching, because they have excess charge carriers in the base region when you drive them to saturation. Baker clamp and Speedup capacitor were tricks used in the past to address this issue. They help, but a MOSFET is just much faster, because it just doesn't have the charge carrier build up problem.
MOSFETs have other challenges, but for this kind of small signal high speed on/off switching they are trivial to use, cheap and readily available. So one answer to the OPs question of "How to reduce transistor switching time" is "Don't. Use a MOSFET".