There are two parts to the puzzle:
Perception speed (how quickly you recognise “incoming”)
Movement speed (how quickly you can move to deal with the threat once you recognise it)
Perception speed is a function of the nervous system. It’s debatable whether it is possible to speed it up in absolute terms, though some nutritional supplements which allegedly improve CNS functioning such as L-Tyrosine, St Johns Wort and acetyl-L-carnitine may help.
More to the point is improving the efficiency of recognising the TYPE of threat to engineer the best response. You can train yourself to be more aware of your surroundings and other people in the street by doing so consciously - crowded places like train stations and shopping malls are good for this.
WC makes much of chi sao to improve sensitivity, but this only works for improving tactile reflexes. No doubt these are useful, if not ideal, once contact has been made, but one also needs to train the visual senses - best done by responding to semi-random and random attacks from out of range. There are certain eye exercises you can do to allegedly improve the efficiency of the muscles controlling the eyeball which may help, but you really want to be able to recognise early when a particular type of attack is coming at you by noticing the movement of the shoulders, hips or limbs, so you know if it’s a straight punch, hook, side kick, roundhouse, etc. as early as possible. Sparring is one example of a random drill.
Most experienced grapplers develop excellent tactile sensitivity, and chi sao hardly has the franchise on this, but stick with the tools of your style if that’s what you have.
For movement speed, as VTS said, look at how sprinters train. These are the fastest humans on the planet. Remember the simplest movements are usually the fastest. Take sprinters’ concepts (working on running technique, moving against resistance, overspeed) and look at how you can adapt them to your punches, kicks, and other techs. Think fast, think loose and snappy. Work on the flow of your combos as well as the speed of single techniques. Often the speed and efficiency which one can sew combos together will be more effective than the ability to throw one fast technique.
Heresy to some perhaps, but boxing tools like spped balls, double end balls and slip bags WILL help.
SLT (and any other form for that matter) is good for learning to do techs with maximum efficiency, which should lead to faster movement speed in time. It will do NOTHING to alter your perception speed, which by definition requires outside stimuli.