| Get lessons, do pair work, etc.
One of the main purposes of fencing exercises is to improve your ability to recognize the situation and react early.
The machine that measures the time it takes you to hit a target after a light comes on is pretty useless since your opponent isn't going to light up when the opportunity to hit him arises.
To improve your reaction time for executing fencing actions, you need to train yourself using fencing actions as the stimulus.
Let's say you want to improve you ability to recognize which line your opponent is attacking. Your coach/partner/android should attack you (and I mean actually try to hit, not some "presenting the blade" crap) to each line in turn, your goal being to parry appropriately. Initially, it's all known and premeditated: you know where your partner is going to attack and which parry you need to take. Further, it's also in place--your partner, of course, lunges with the attack, but there is no maneuvering before that. Also--and this is possibly the most important thing--the attacks should not come at rhythmic intervals. Though you know where the attack is going, you don't know when your partner will execute his attack. This allows you to develop your ability to react to the movement, rather than the rhythm.
Next, do the same thing without premeditating which attack will come next. In other words, your partner will make attacks to any line he wishes, your goal will be to identify which parry to take. Your partner should be making lunges and actually trying to hit you. At first he may be lunging slowly, but it should be a real movement nonetheless. Again, he must not be attacking rhythmically. The intervals between attacks must always vary.
Next, do the same thing, but while maneuvering. So you're both moving on the strip as you would in a bout (meaning no one person is designated "leader" of movement: you're both trying to control the movement) and your partner makes attacks at different intervals to different lines, while you have to choose the appropriate parry in time.
Again, the attacks may initially be somewhat slower than usual, but the movement must be real--if you choose the wrong parry, you should be hit--as you must train yourself to recognize the movement of someone actually trying to hit you.
This, of course, is one example (choosing the right parry by identifying the line of attack in time), but the same methods can be applied to any reaction in fencing. Choose some reactions that appear in bouts (attacking on your opponent's step forward, feinting into an invitation, counterattacking against an opponent who is rushing, etc.) and first practice doing them as isolated actions on your partner's signal, then together (choosing which action to use based on your partner's realistic movement), then with mobility. And, again, always be sure to eliminate rhythmic execution from the exercise.
It'll also be important that your partner try to eliminate any presignal from his movement. Of course, part of what can make your reaction time improve significantly is your improved ability to recognize a presignal, but your partner should be as smooth as possible as a quality opponent would also be. Your ability to recognize subtle presignals will then improve. | 
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